Category: Maths Notes PPT

In our day-to-day life, we came across various solids having different shapes and size, in which we can calculate both surface area and volume. We need to calculate the surface area and volume of the objects around us. But what if these basic shapes combine and form some different shape other than the original one. Now, the question is how you will calculate the volume, surface, and areas of new objects. While calculating the surface area and volumes of these new shapes, we need to observe the new form. A deep discussion is given below that will create a more clear image of these objects and their calculation. 

 

Solid made up of common geometric solids is known as composite solid.  They are made up of pyramids, prisms, cylinders, pairs, and cones to find out the volume and surface area of a solid. We need to find out the volume and surface area of prisms,  cons, spheres, pyramids, and cylinders. Sum total of the surface areas of the individual solids that make up the combined solid excluding the overlapping parts from each figure is the total area of a combined solid whereas the sum of the volumes of individual solids that make up the combined solid is the volume of a combined solid.

Combination of Solids – Surface Area

We should only calculate the areas that are visible to our eyes while calculating the surface area of the combined solid. For example, we need to just find out the C.S.A. of the hemisphere and of the cone separately and sum them up together if a cone is surmounted by a hemisphere.

The surface area of an object is given by the total area of the surface that an object occupies, or we can say the total area of all the surfaces of any three-dimensional figure. The surface area of figures other than cubes or cuboids can be calculated as the lateral area of the figure plus its every base, in case if prism and cylinder are the same then we can take it as twice the area of the base. The surface area of any given figure can be calculated with the help of the example of a gift as a three-dimensional figure and let the surface area be the wrapping paper, so the amount of wrapping paper used to cover the gift is the surface area of the given three-dimensional figure. Surface area can be given by the following formula:

 

Surface area = Lateral area + (n * base)

 

where, n = no. of bases present (n = 2 for prism/ cylinder, n = 1 for pyramids/ cones, and n = 0 for spheres/ circles)

 

Surface area can be further divided into 2 types such as:

1. Total surface area – The area including the base and the curved part is called a total surface area.

2. Curved surface area – The area of the curved part excluding the base is called the curved surface area.
   

Volumes of any given object can be said as the amount of liquid it can contain in it. Basically, the quantity enclosed by the given three-dimensional objects is called the volume of that object. The volume of the one dimensional (e.g., lines), as well as the two-dimensional object (e.g., squares), are considered zero as the volume is considered as quantity. The basic properties to find the volume of any given object are as follows:

1. Any given object has the volume of length * breadth * height (V = lwh).

2. The total volume of any given object is the sum of all non-overlapping regions.

3. Exactly the same when superimposing figures have the same volume.

4. Depending on the unit cube, every polyhedral region has a unique volume.
   

Volume can be calculated with the help of the following formulas for different figures:

Volume of the sphere (V) = (4/3) π x (radius)³

Volume of prism or cylinder (V) = base area x height

Volume of pyramid or cone (V) = [(frac {1}{3})] x base area x height

Combination of solids

We came across different shapes which are the combination of different solids having one or more basic shapes. Some of the most common examples of the combination of solids include ice-cream cones, capsules, tents, and capsule-shaped trucks carrying petrol or liquefied petroleum gas. Basically, the combination of solids can be explained as those solids which can break into two or more different solids having sides. The combination of solids is also known as composite shapes as the combination of solids formed by the fusion of two or more different shapes to form a new shape.

 

To calculate the surface area or the volume of these types of solids, first, we have to see the number of solid shapes that form these shapes, as these three-dimensional structures contain various one-dimensional shapes having an example of a cube formed with the help of six squares which is a one-dimensional shape. The surface area of a given composite shape is the sum of the area of all the faces in that solid. To understand the combination of solids, we can take one example of ice cream-filled cone, which is the fusion of a cone and the hemisphere-shaped ice cream. So, the total surface area of the ice cream-filled cone is equal to the sum of the curved surface area of the hemisphere and the curved surface area of the cone.

 

The curved surface area of cone = πrl,

And the curved surface area of hemisphere = [pi r^2]

So, the total surface area of ice cream filled cone = [pi r^2 + pi rl]

 

To calculate the volume of the combinational shapes, we have to first figure out the different shapes involved in it to form the composite shape. The volume of the combinational shapes can be calculated by calculating the volume of the specific shapes through which a new combinational shape is formed and adding them to form the total volume of the composite shape.

 

Similarly, in the case of volume, the volume of the ice cream-filled cone can be found out by individually calculating the volume of the hemisphere and the volume of the cone and then adding up to form the volume of the ice cream-filled cone.

 

Volume of cone = [frac {1}{3}] πr²h

 

Volume of hemisphere = [frac {2}{3}] πr³

 

Important formulas of surface area and volume are as below:

Linear and quadratic equations are the algebraic systems of equations consisting of one linear equation and one quadratic equation. The goal of solving systems linear quadratic equations is to significantly reduce two equations having two variables down to a single equation with only one variable. Since each equation in the system consists of two variables, one way to decrease the number of variables in an equation is by substituting an expression for a variable.

In a test, you will be expected to identify the solution(s) to systems either algebraically or graphically.

Examples of Linear Equation and Quadratic Equation

Following are the example of Linear and quadratic equations:

y = x + 1

y = x² −1

In systems linear quadratic equations, both equations are simultaneously true. Simply to say, because the 1st equation tells us that y is equivalent to x + 1, the y in the 2nd equation is also equivalent to x + 1. Hence, we can plug in x + 1 as a substitute for y in the 2nd equation:

y = x² − 1

x + 1 = x² −1

From here, we are able to solve and simplify the quadratic equation for x, which provides us with the x-values of the solutions to the linear-quadratic system. Then, we can also use the x-values and either equation in the system in order to find out the y-values.

Solving Linear-Quadratic Systems

You might have solved systems of linear equations. But what about a system of 2 equations where 1 equation is linear and the remaining is quadratic?

We can apply a version of the substitution technique in order to solve systems of this type.

Note that the standard form of the equation for a parabola with a vertical axis of symmetry is y = ax² + bx + c, a ≠ 0 and the slope-intercept form of the equation for a line is y = mx + b,

To avoid any sort of confusion with the variables, write the linear equation as y= mx+d where,

m = The slope.

d = The y-intercept of the line.

Substitute the expression for y from the linear equation, into the quadratic equation. In other words, substitute mx + d for y in y = ax² + bx + c.

mx + d = ax² + bx + c

Now, rewrite the new quadratic equation in the standard form.

Subtract mx+d from both sides of the equation

(mx + d) − (mx + d) = (ax² + bx + c) − (mx + d)0 = ax² + (b − m)x + (c − d)

Now we have a quadratic equation in 1 variable, the solution of which can be determined with the help of the quadratic formula.

The solutions to the equation ax² + (b − m)x + (c − d) = 0 will provide us with the x-coordinates of the points of intersection of the graphs of the parabola and the line. The corresponding y-coordinates can be identified with the help of the linear equation.

Another way of solving the system is to graph the two functions on a similar coordinate plane and then determine the points of intersection.

Solved Examples

Example:

Determine the points of bisection between the line y = 2x + 1 and the parabola y = x² − 2.

Solution:

Substitute the values in the equation 2x + 1 for y in y = x² − 2.

2x + 1 = x² − 2

Now, express the quadratic equation in standard form.

2x + 1− 2x − 1 = x² − 2 − 2x − 10 = x² − 2x − 3

Apply the quadratic formula in order to identify the roots of the quadratic equation.

Here,

a = 1

b = −2

c = −3

x = √[− (−2) ± (−2)2 − 4(1)(−3)]/2(1)

= √[2 ± 4 + 12]/2

= 2 ± 4/2

= 3, −1

Then, Substitute the x-values in the linear equation in order to identify the corresponding y-values.

x = 3 ⇒ y = 2(3) + 1 = 7

x = −1 ⇒ y = 2(−1) + 1 = −1

Thus, the points of bisection are (3, 7) and (−1, −1).

Also, remember that the same method can be used to determine the intersection points of a line and a circle.

Check below the Graphing of the parabola and the straight line on a coordinate plane.

Example: Determine the points of intersection between the line y = −3x and the circle x² + y² = 3.

Solution:

Substitute −3x for y in x² + y² = 3.

x² + (−3x)² = 3

Simplify the system of quadratic equations

x² + 9x² = 3

10x² = 3

x² = 3/10

Taking square roots, x = ±√(3/10).

Now, Substituting the x-values in the linear equation in order to identify the corresponding y-values.

x = √(3/10) ⇒ y = −3(√(3/10))

=−(3/3√10)

x = −√(3/10) ⇒ y =−3(−√(3/10))

= 3/3√10

Hence, the points of intersection come out to be (√(3/10),  (-3√3/10), and (−3/√10√,  3√3/√10).

Refer below for the Graphing of the circle and the straight line on a coordinate plane.

Fun Facts

• While solving quadratic simultaneous equations, if we get a negative number as the square of a number in the outcome, then the two equations do not have real solutions.

• A  quadratic and linear system can be represented by a line and a parabola in the xy-plane.

• Each intersection of the line and the parabola depicts a solution to a linear-quadratic system.

What is Test of Significance?

A test of significance may be a formal procedure for comparing observed data with a claim (also called a hypothesis), the reality of which is being assessed. It may be a statement with a few of the parameters, like the population proportion p or the population mean µ.

Once the sample data has been collected through an observational study or an experiment, statistical inference will allow the analysts to assess the evidence in favor or some claim about the population from which the sample has been taken from. 

Null Hypothesis

Every test for significance starts with a null hypothesis H0. H0 represents a theory that has been suggested, either because it’s believed to be true or because it’s to be used as a basis for argument, but has not been proved. For example, during a clinical test of a replacement drug, the null hypothesis could be that the new drug is not any better, on average than the present drug. We would write H0: there’s no difference between the 2 drugs on average.

Alternative Hypothesis

The alternative hypothesis, Ha, maybe a statement of what a statistical hypothesis test is about up to determine. For example, during a clinical test of a replacement drug, the choice hypothesis could be that the new drug features a different effect, on the average, compared to the current drug. We would write Ha: the 2 drugs have different effects, on the average. The alternative hypothesis may additionally be that the new drug is better, on the average than the present drug. In this case, we might write Ha: the new drug is better than the present drug, on the average.

The final conclusion once the test has been administered is usually given in terms of the null hypothesis. Either we “reject the H0 in favor of Ha” or “we do not reject the H0“; we never conclude “reject Ha“, or maybe “accept Ha“.

What is Test of Significance?

Two questions come up about any of the hypothesized relationships between the two variables:

1) what’s the probability that the connection exists;

2) if it does, how strong is that the relationship

There are two sorts of tools that are required to address these questions: the primary is addressed by tests for statistical significance; and therefore the second is addressed by Measures of Association.

Tests for statistical significance are required to address the question: what’s the probability that what we expect may be a relationship between two variables is basically just an opportunity occurrence?

If we select many samples from an equivalent population, would we still find an equivalent relationship between these two variables in every sample? Suppose we could do a census for the population,  we will also find that this relationship exists within the population from which the sample was taken from? Or will it be our finding due to only random chances?

Let’s Know What Tests for Statistical Significance Tell Us.

Tests for statistical significance tell us what the probability is that the relationship we expected we’ve found is due only to random chance. They tell us what the probability is that we might be making a mistake if we assume that we’ve found that a relationship exists.

We can never be 100% sure that the relationship always exists between two variables. There are too many sources of error to be controlled, for instance , sampling error, researcher bias, problems with reliability and validity, simple mistakes, etc.

But using applied Mathematics and therefore the bell-shaped curve, we will estimate the probability of being wrong, if we assume that our finding a relationship is true. If the probability of being wrong is very small, then our observation of the connection can also be a statistically significant discovery.

Statistical significance means there’s an honest chance that we are right to find that a relationship exists between two variables. But the statistical significance isn’t equivalent as practical significance. We can have a statistically significant finding, but the implications of that finding may dont have any application. The researcher should examine both the statistical and therefore the practical significance of any research finding.

Test of Significance in Statistics

Technically speaking, in the test of significance the statistical significance refers to the probability of the results of some statistical tests or research occurring accidentally. The main purpose of performing statistical research is essential to seek out reality. In this process, the researcher has to confirm the standard of the sample, accuracy, and good measures that require a variety of steps to be done. The researcher determines whether the findings of experiments have occurred thanks to an honest study or simply by fluke.

 

The significance may be a number that represents probability indicating the results of some study has occurred purely accidentally. The statistical significance may be weak or strong. It does not necessarily indicate practical significance. Sometimes, when a researcher doesn’t carefully make use of language within the report of their experiment, the importance could also be misinterpreted.

 

The psychologists and statisticians search for a 5% probability or less which suggests 5% of results occur thanks to chance. This also indicates that there’s a 95% chance of results occurring NOT accidentally. Whenever it’s found that the results of our experiment are statistically significant, it refers that we should always be 95% sure the results aren’t due to chance.

 

Process of Significance Testing in Test of Significance

So in this process of testing for statistical significance, the following are the steps:

1. Stating a Hypothesis for Research

2. Stating a Null Hypothesis

3. Selecting a Probability of Error Level

4. Selecting and Computing a Statistical Significance Test

5. Interpreting the results

A transformation is that geometric process of mathematics that maneuvers a polygon, quadrant or other 2-D object on a plane or coordinates system. These transformation maths helps to explain how two-dimensional figures move around about a coordinate plane. Vectors are most commonly used to describe translations in mathematics. Thus, in simple terms, a translation of an image takes place when a shape is moved from one place to another. It is just like gathering up the shape from its original place and putting it down somewhere else. That being said, any object or image in a plane or a coordinate system could be manipulated by utilizing different operations of transformations.

Two – Dimensional Shapes

A preimage or an inverse image is the 2D shape that we used to denote before any transformation. Image is the figure that we get after transformation.

Types of Transformations

We have 5 different transformations or operations of transformations in math which are as follows:

1. Reflection – The image is a mirrored preimage; what we mathematically call a “a flip.”

2. Rotation – The image is the preimage rotated around a specified point; what we mathematically call “a turn.”

3. Dilation – The image is a smaller or a larger version of the preimage; what we mathematically call a “shrinking” or “enlarging.”

4. Shear – All the points across one side of a preimage does not change (remain fixed) while all other points of the preimage move parallel to that side in distribution to the distance from the given side; what we mathematically call “a skew.,”

5. Translation – The image is equalized by a constant value from the preimage; what we mathematically call “a slide.”

Now that you are aware of the different transformations that can occur, let us learn about them in detail as to appropriately use them further.

Reflection

Imagine snipping off a preimage, picking it up, and putting it down again at face. That is what the process of a reflection or a flip is. You already know that a reflection image is an exact duplicate of the preimage. So, now tell us if the below  triangle image in purple is a reflection of the purple preimage or object?

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Since the purple triangle image is being reflected through the x-axis, it is a reflection image. Here, you also need NOT to use a coordinate plane’s axis for to identify a reflection.

Rotation

A two-dimensional figure does not have to be dependent on the origin for rotation. You can easily see rotation in these figures using the coordinate grid which is as easy as using the x-axis and y-axis. For example,

In order to create rotation in 90° 90°: (x, y) → (−y, x)(x, y) → (-y, x) (multiply the y-value times −1-1 and shift the x- and y-values)

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Dilation

Dilate a preimage of any polygon is performed by making a carbon copy of its interior angles while increasing every side in proportion. Imagine dilating as resizing and you will master over the concept of dilation geometry. Below is the image green which is a dilation of the purple preimage. 

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Shear

To shear an image, you “skew it,” meaning that fixing one line of the figure whereas moving all the other lines & points in a specified proportion, direction to their distance without changing its area, interior angles or stretching dimensions.

Below you will find a square figure transformed into a parallelogram without changing dimensions and angles. 

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Translation

A translation moves the object from its native place on the coordinate grid without having to alter its orientation. 

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Solved Examples

Translation Transformation Example 1

If you are asked to translate a figure into a coordinate plane, how will you do it?

For translating a polygon MNO on the coordinate plane, your mathematical graphing instructions look like this:

Taking a Triangle MNO + 9 units, we will translate the triangle on the coordinate plane in the x-direction and -4 in the Y direction.

Using Method by Writing down the original coordinates,

                 X           Y

M (-8, 6) — 8 + 9   6 – 4

N (-8, 9) — 8 + 9         9 – 4

O (-4, 6) — 4 + 9         6 – 4

Thus, we get

M (1, 2)

N (1, 5)

O (5, 2)

Hence, translated polygon should be congruent to the original

Translation Transformation Example 2

Problem

Determine the components of the translation, and identify where P point’ ends up:

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Solution

To identify the translated vector, we need to choose a labeled point and then subtract the old coordinates from the new. That is to say:-

Translation

A'(-1, 2) – A (1, 4)

(-2, -2)

P’ = P (-1, 3) + (-2, -2)

Your answer is (-3, 1)

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Fun Facts

• You can easily Rotate 90 ° Clockwise or 270 ° Counterclockwise About the Origin using the mathematical formula (x, y) ⟶ (y, -x).

• Reflecting a polygon along x-axis means measuring the distance of each vertex to the line of reflection.

• Perfect examples of shear are Italic letters on a computer.

Triangle is a polygon or shape that has three sides that meet at a point. A triangle has three angles and three points. There are different types of triangles and each one has different characteristics and meaning.

The first type of triangle we will discuss is the right triangle. A right triangle is a triangle with a right angle. A right angle is an angle that measures exactly 90 degrees. One of the most common right triangles is a 45-45-90 triangle. This triangle is named as such as one of the angles measures 45 degrees and the other two measure 45 degrees as well.

Another type of triangle is the obtuse triangle, meaning that one of the angles is greater than 90 degrees. An example of an obtuse triangle is one that is shaped like the letter L. An isosceles triangle is a triangle with at least two equal sides and equal angles. An equilateral triangle has three equal sides and three equal angles and is the only triangle where the angles and sides all measure the same.

Importance of Studying Triangles

Triangles are important to study because they can be used to study certain formulas that measure the areas and the lengths of the sides of a triangle. Triangles can also be used to measure the area of a quadrilateral. Take a look at a few of the many ways to study triangles.

The following methods to study triangles are efficacious for finding the basics of triangles and how to measure the different parts of the triangle.

The first way of studying triangles is to read about them. Reading about triangles can show you the definitions of triangles and how they work, which is the most important thing to know.

The second way is to use visual aids. These can include drawings or actual triangles made out of foam board or rubber matting.

The third way of studying triangles is to build a triangle and measure the different parts such as the height, base, and the angle which forms the side opposite of the base. This will be helpful to the student because they can visually see how the parts work and what part is opposite of each.

The fourth way is to build a triangle- It is important to build a triangle out of a rubber band so you can see the shape and measure the dimensions.

The last way is to Learn the rules- It is important to learn the general rule that when studying a triangle that the area of the triangle is equal to one half the product of the base and height, and when measuring a quadrilateral, the area is equal to one half the product of the base and the height.

Maths Class 9 Triangles

• We have learned different types of 2-dimensional figures like square, rectangle, triangle, circle, etc. and their various properties. In triangles class 9 notes, we will be discussing the figure triangle in detail.

• A figure formed by the intersection of three lines is said to be a triangle. A triangle has three vertices, three sides, and three angles.

• The above figure shows ABC, here AB, BC, AC are the sides of the triangle. A, B, C are the vertex and ∠ A, ∠B, ∠ C are the three angles.

• We have studied the different properties of triangles. Now, let us learn different types of triangles, the congruence of triangles, and the inequality relations of triangles.

• Triangles class 9 notes will help you to solve problems related to triangles easily.

Congruent Triangles

Two geometrical figures having exactly the same shape and size are said to be congruent figures.

Two triangles are congruent to each other if one of them is superimposed on another such that they both cover each other completely.

Two triangles are said to be congruent if the sides and angles of one triangle are exactly equal to the corresponding sides and angles of the other triangle.

From the figure, ABC is congruent to DEF, and it is written as ABC   ≅  DEF

In two congruent triangles, corresponding parts of corresponding angles generally written as ‘c.p.c.t’ are equal they are:

∠ A = ∠ D, 

∠ B = ∠ E,

∠ C = ∠ F  

and

AB = DE ,

BC = EF ,

AC = DF.

Note:

Every triangle is congruent to itself, i.e. ABC   ≅  ABC

If ABC   ≅  DEF, then DEF   ≅  ABC

If ABC   ≅  DEF and DEF   ≅ PQR then ABC   ≅ PQR

Criteria for Congruence of Triangles 

Criteria for the congruence of triangles are well defined and proved. Congruent parts of the congruent triangle are written as c.p.c.t. Different rules for congruence are as given below:

SAS (Side-Angle-Side) Congruence Rule

Suppose two sides and the angle included between the two sides of one triangle are equal to the corresponding sides and the included angle of the other triangle. In that case, the two triangles are congruent to each other.

In the above figure  AB = PQ ; 

AC = PR; 

And the angles between the sides are equal.

I.e ∠ A = ∠ P 

therefore ABC   ≅ PQR …….by SAS criteria

also,∠ B = ∠ Q; ∠ C = ∠ R; BC  = QR (by c.p.c.t)

ASA (Angle-Side-Angle) Congruence Rule

Suppose two angles and the side included by the two angles of one triangle are equal to the corresponding angles and sides included by the angles of the other triangles. In that case, the two triangles are said to be congruent. 

In the above figure  ∠ B = ∠ Q 

∠ C = ∠ R 

And the side between the angles are equal

BC = QR;

therefore ABC   ≅ PQR …….by ASA criteria

also,∠ A = ∠ P; AB = PQ ; AC  = PR (by c.p.c.t)

AAS (Angle-Angle-Side) Congruence Rule

If two angles and one non-included side of one triangle are equal to corresponding angles and a non-included side of another triangle then the two triangles are said to be congruent to each other.

In the above figure  ∠ B = ∠ E 

∠ C = ∠ F 

and

AC = DF;

Then  ABC   ≅ DEF …….by AAS criteria

also,∠ A = ∠ D; AB = PQ ; BC  = QR (by c.p.c.t)

SSS (Side-Side-Side) Congruence Rule

If all the three sides of a triangle are equal to the corresponding sides of another triangle then the two triangles are said to be congruent to each other.

In the above figure  AB = PQ ; 

BC = QR; and

AC = PR, 

therefore ABC   ≅ PQR …….by SSS criteria

Also,∠ A =
∠ D; ∠ B = ∠ E; ∠ C = ∠ F (by c.p.c.t)

RHS (Right angle-Hypotenuse-Side) Congruence Rule

Two right-angled triangles are congruent if one side and the hypotenuse of the one triangle are equal to the corresponding side and the hypotenuse of the other.

In the above figure, hypotenuse XZ = RT

And side YZ=ST, 

∠ XYZ = ∠ RST ( angle are of 900) 

therefore XYZ   ≅ RST …….by RHS criteria

Also,∠ X = ∠ R; ∠ Z = ∠ T;  XY = RS (by c.p.c.t)

Inequality Relations In a Triangle

If two sides of a triangle are unequal, the longer side has a greater angle opposite to it, here if in ABC, BC > AB. then ∠ CAB > ∠ ACB 

The Triangle Inequality theorem states that

The sum of the lengths of any two sides of a triangle is greater than the length of the third side of a triangle.

Let us see some triangle questions for class 9

Solved Examples

Example 1: In the below figure AD = BC and ∠DAB = ∠ CBA.Prove that AC = BD and ∠BAC = ∠ABD

Solution: In DAB and CBA

AD  = BC…..       (Given)

∠DAB = ∠ CBA….  (Given)

AB =AB….. (common side)

So by SAS criteria of congruence we get,

DAB ≅ CBA

So by corresponding parts of congruent triangle

AC = BD and ∠BAC = ∠ABD

Example 2: In the right triangle ABC, the right angle at C, M is the midpoint of hypotenuse AB. Join CM and produce to a point D such that DM = CM. Point D is joined to point B(see figure). Show that:AMC ≅ BMD

Solution: In AMC and  BMD

AM = MB ( M is the midpoint of AB given)

∠DMB = ∠ CMA ( vertically opposite angles)

CM = MD  ( given )

therefore , by SAS criteria , we get

AMC ≅ BMD hence proved

Solve more problem triangles lessons for class 9 and be an expert at solving triangles problems.

Triangle Questions for Class 9

In the given figure, ΔABD and ΔACE are equilateral triangles that are drawn on the sides of a ΔABC. Prove that CD = BE.

In the given figure, side AB = AD, side AC = AE and ∠BAD = ∠EAC, then prove that BC = DE.

Trigonometric values of ratios like sine, cos, tan, cosec, cot, and secant are very useful while solving and dealing with problems related to the measurement of length and angles of a right-angled triangle. 0°, 30°, 45°, 60°, and 90° are the commonly used values of the trigonometric function to solve trigonometric problems.

The concept of trigonometric functions and values is one of the most important parts of Mathematics and also in our day-to-day life.

Trigonometric Ratios

Trigonometry values are based on three major trigonometric ratios, Sine, Cosine, and Tangent.

• Sine or sin θ = Side opposite to θ / Hypotenuse = BC / AC

• Cosines or cos θ = Adjacent side to θ / Hypotenuse = AB / AC

• Tangent or tan θ =Side opposite to θ / Adjacent side to θ = BC / AB

Similarly, we will write the trigonometric values for reciprocal properties, Sec, Cosec and Cot ratios.

• Sec θ = 1/Cos θ = Hypotenuse / Adjacent side to angle θ = AC / AB

• Cosec θ = 1/Sin θ = Hypotenuse / Side opposite to angle θ = AC / BC

• Cot θ = 1/tan θ = Adjacent side to angle θ / Side opposite to angleθ = AB / BC

Also,

• Sec θ . Cos θ =1

• Cosec θ . Sin θ =1

• Cot θ . Tan θ =1

Values of Trigonometric Ratios

Sign of Trigonometric Functions

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Trigonometry Ratios Formula

• Tan θ = sin θ/cos θ

• Cot θ = cos θ/sin θ

• Sin θ = tan θ/cos θ

• Cos θ = sin θ/tan θ

• Sec θ = tan θ/sin θ

• Cosec θ = cos θ/tan θ

Also,

• sin (90°- θ) = cos θ

• cos (90°- θ) = sin θ

• tan (90°- θ) = cot θ

• cot (90°- θ) = tan θ

• sec (90°- θ) = cosec θ

• cosec (90°- θ) = sec θ

• sin (90°+ θ) = cos θ

• cos (90°+ θ) = -sin θ

• tan (90°+ θ) = -cot θ

• cot (90°+ θ) = -tan θ

• sec (90°+ θ) = -cosec θ

• cosec (90°+ θ) = sec θ

• sin (180°- θ) = sin θ

• cos (180°- θ) = -cos θ

• tan (180°- θ) = -tan θ

• cot (180°- θ) = -cot θ

• sec (180°- θ) = -sec θ

• cosec (180°- θ) = cosec θ

• sin (180°+ θ) = -sin θ

• cos (180°+ θ) = -cos θ

• tan (180°+ θ) = tan θ

• cot (180°+ θ) = cot θ

• sec  (180°+ θ) = -sec θ

• cosec (180°+ θ) = -cosec θ

• sin (360°- θ) = -sin θ

• cos (360°- θ) = cos θ

• tan (360°- θ) = -tan θ

• cot (360°- θ) = -cot θ

• sec (360°- θ) = sec θ

• cos
  ec (360°- θ) = -cosec θ

• sin (360°+ θ) = sin θ

• cos (360°+ θ) = cos θ

• tan (360°+ θ) = -tan θ

• cot (360°+ θ) = -cot θ

• sec (360°+ θ) = sec θ

• cosec (360°+ θ) = -cosec θ

• sin (270°- θ) = -cos θ

• cos (270°- θ) = -sin θ

• sin (270°+ θ) = -cos θ

• cos (270°+ θ) = sin θ

• Tan θ = sin θ/cos θ

• Cot θ = cos θ/sin θ

• Sin θ = tan θ/cos θ

• Cos θ = sin θ/tan θ

• Sec θ = tan θ/sin θ

• Cosec θ = cos θ/tan θ

From that we can say that, 

• Sec θ . Cos θ =1

• Cosec θ . Sin θ =1

• Cot θ. Tan θ =1

The trigonometric tables are basically provided to list down the ratio of angles such as 0°, 30°, 45°, 60°, 90° as well as angles such as 180°, 270°, 360°. It is a tabular summary of the values. Predicting the values ​​in the trigonometric tables and using that table as a reference to calculate the values ​​of trigonometric functions at various other angles based on the patterns found within the trigonometric ratios and even between angles becomes easy while solving problems.

The sine function, cosine function, tan function, cot function, sec function, and cosine function are all trigonometric functions. You can use the trigonometric tables to find the angle values ​​for standard trigonometric functions such as 0^∘, 30^∘, 45^∘, 60^∘, 90^∘. There are various trigonometric ratios such as sine, cosine, tangent, cotangent, second, cotangent, and so on. According to mathematical criteria, sin, cos, tan, cosec, sec, and cot are abbreviations for these ratios. To solve the trigonometry problem, students need to remember these standard values.

The trigonometric tables are a collection of standard angle trigonometric ratio values, including 0°, 30°, 45°, 60°, and 90°. It may also be used to find values ​​for other angles such as 180°, 270°, 360° in the form of a table. You can notice different patterns within the trigonometric ratios and between each angle, knowing those can make it a lot easier to solve the problems quickly. Therefore, it is easy to predict the values ​​in the trigonometric tables, and you can also use the table as a reference for calculating trigonometric values ​​for other non-standard angles. The various trigonometric functions in mathematics are sine, cos, tan, cot, sec, and cosec functions. 

Tricks to Remember the Trigonometric Values 

1. One easy way to remember the values is to just learn the sin values first. Memorize the sin values from 0° – 90° and then the cos values are just the backward values of sin, i.e. from sin 90° – 0° are the values of Cos 0° – 90°. 

After that follow the formulas, such as Tan value is sin/cos, Cosec value is inverse of sin value, Sec value is inverse of Cos values and Cot value is inverse of tan value. 

2. For Sin values, Count the fingers on your hand from 0-4 from left to right and consider them as angles from 0° – 90°. Divide each of them by 4 ( 0/4, 1/4…)  and take the square root of the values( 0, 1/2…) to get the required angles Except for angle 45°, which is obtained by taking the square root of the previous angle, ie. of angle 30°.