[Maths Class Notes] on Sin 45 Degree Pdf for Exam

Sine is considered one of the most important functions in trigonometry as it is used to find out the unknown values of the angles and length of the sides of a right-angled triangle. It is necessary to talk about the importance of Sine functions in trigonometry before having a discussion on Sin 45 degrees. In a right-angled triangle, the Sine function can be defined as the ratio of the length of the opposite side of a triangle to its hypotenuse. We generally require the value of Sin 45 degrees and the value of other trigonometry ratios for all the degrees such as 0, 30°,45°,60°,90°,180° to solve the questions based on trigonometry.

We will discuss Sin 45 value in this article.

Importance of Sine and Other Trigonometric Functions

In the general population of non-mathematicians and non-scientists, trigonometry is mainly thought to be useful in measurement problems. But many remain unaware of how trigonometry silently seeps into their daily lives as well. For example, in the theory of music.

In other cases, the use is still more technical, such as in the number theory. Many mathematical concepts such as the Fourier series and Fourier transforms are heavily dependent on the knowledge of trigonometric functions that find application in a number of areas including statistics.

Traditionally, many scientific fields mentioned make use of trigonometry. They are acoustics, architecture, astronomy, cartography, civil engineering, geophysics, crystallography, electrical engineering, electronics, land surveying and geodesy, many physical sciences, mechanical engineering, machining, medical imaging, number theory, oceanography, optics, pharmacology, probability theory, seismology, statistics and visual perception. 

Great Trigonometrical Survey

The Great Trigonometrical Survey was a massive project undertaken during the years 1802 up until 1871 to survey the Indian subcontinent with high precision. Mathematicians and geographers triangulated vast distances across the country starting from the coastal baseline. One of the key achievements of this survey was that for the first time they were able to scale the height of the Himalayan mountains and conclude that Mount Everest is the highest point on the Earth.

One important thing to note is that in some of the classical fields of mathematical sciences such as navigation and land surveying, the use of trigonometry is obvious. However, in other non-classical cases, such as the development of music theory, the application of trigonometry is hidden. Yet some musicians might know how it is related to work begun by Pythagoras, who observed that the sounds made by plucking two strings of different lengths are consonant in the case when both lengths are small integer multiples of a common length. The similarities between the shape of a vibrating string, the shapes of some waves, etc., and the graph of the sine function is not merely coincidental. Rather, these are specific studies originating from the knowledge of periodicity of sine and cosine functions.

Sin 45 Value

Let us take a right angle triangle Δ ABC to know about the Sin 45 value. The Sin ϴ of an angle can be defined as the ratio of the length of the opposite side of a triangle to its hypotenuse.

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Based on the above diagram:

Sin ϴ = Opposite side or perpendicular / Hypotenuse

Sin ϴ = AC/AB

Sin ϴ  =  B/C

We will discuss here a simple procedure through which you can easily find out the Sin 45 value and the value of other Sin ratios with different degrees.  If you understand this method thoroughly, you will be able to calculate the values of all the trigonometric ratios which are used to solve trigonometry equations.

Calculation of the Value of Sin 45 Degree in Fraction

• Sin 0° = [frac{sqrt{0}}{sqrt{4}}] = 0

• Sin 30° = [frac{sqrt{1}}{sqrt{4}}] =1/2

• Sin 45° = [frac{sqrt{2}}{sqrt{4}}] = [frac{1}{sqrt{2}}]

• Sin 60° = [frac{sqrt{3}}{sqrt{4}}] = [frac{sqrt{3}}{2}]

• Sin 90° = [frac{sqrt{4}}{sqrt{4}}] = 1

Similarly, the values of other trigonometric ratios such as cosine and tangent can be found out.

Now, we will learn sine ratios for the different degrees based on the table given below.

Sine Ratio Table

Q. What is the exact value of Sin 45° geometrically?

Ans: Let us find the exact value of Sin 45 using an isosceles triangle.

An isosceles triangle with perpendicular side =1 cm and hypotenuse = √2 will give you the value of Sin 45 degrees.

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As we know,

Sin 45° = Perpendicular/ hypotenuse

Sin 45 ° = AB/AC

Sin 45° =  [frac{1}{sqrt{2}}]

So, the exact value of Sin 45 degree is [frac{1}{sqrt{2}}]

The exact value of Sin 45 degree in decimal form is 0.7071067812.

Here are some Sine Formulas according to Compound Angles:

[Sin ( alpha + beta ) = Sin alpha + Cos beta +  Cos beta + Sin beta ]

[Sin ( alpha – beta ) = Sin alpha – Cos beta –  Cos beta – Sin beta ]

Solved Examples

1. Evaluate the following:

1. 2 Sin 30° + 3 Cos 60° – 3 Tan 45°

Ans: Value of Sin 30° =[frac{1}{2}]

Value of Cos 60° = [frac{1}{2}]

 Value of tan 45° = 1

Substituting the Values,

= 2(1/2) + 3(1/2)- 3(1)

= 1+ 3/2 – 3

= – 1/2

2. 3( Cos 30°)[^{2}] + 2( Sin 30°)[^{2}]

Ans: Value of Cos 30° = [frac{sqrt{3}}{2}]

Value of Sin °30 =  [frac{1}{2}]

Substituting the values,

=[3(frac{sqrt{3}}{2})^{2} + 2(frac{1}{2})^{2}]

=3(3/4) + 2(1/4)

= 9/4 + 2/4

= 11/4

3. Sin (45°+30)=  [frac{sqrt{2}(sqrt{3}+1)}{4}]

Sin (α+ B) = Sin α + Cos β+ Cos β + Sin β

Ans: Substituting the values using the above formula, we get 

Sin (45°+30) = Sin 45° Cos 30° + Cos 45° Sin 30°

= Sin (45°+30° ) = [frac{sqrt{2}}{2} times  frac{sqrt{3}}{2} + frac{sqrt{2}}{2} times frac{1}{2}]

Taking [sqrt{2}] as common, we get

Sin (45°+30° ) =   [frac{sqrt{2}(sqrt{3}+1)}{4}]

Hence, proved.

Fun Fact

• The 8 trigonometric identities used in trigonometry are also known as fundamental identities. Among 8 trigonometry identities, 3 of the identities are known as Pythagoras identities as they are based on the Pythagorean Theorem.

• Babylonians, the Egyptians based trigonometry on their base 60 numeral system.

• Trigonometry is primarily associated with architecture and music.

• The two most important trigonometric functions which are widely used are Sine and Cosine.

Quiz Time

1. The two trigonometric functions such as Sine and Cosine are different by the addition of the prefix “co” to “cosine”. From where does the word “co” derive?

1. Complementary

2. Constraint

3. Coefficient

4. Constant

2. The sine of an angle ϴ is equivalent to

1. Opposite side divided by the adjacent side

2. The opposite side is divided by the hypotenuse

3. Adjacent side divided by the hypotenuse

4. Hypotenuse divided by the adjacent side