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.
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Sine or sin θ = Side opposite to θ / Hypotenuse = BC / AC
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Cosines or cos θ = Adjacent side to θ / Hypotenuse = AB / AC
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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.
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Sec θ = 1/Cos θ = Hypotenuse / Adjacent side to angle θ = AC / AB
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Cosec θ = 1/Sin θ = Hypotenuse / Side opposite to angle θ = AC / BC
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Cot θ = 1/tan θ = Adjacent side to angle θ / Side opposite to angleθ = AB / BC
Also,
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Sec θ . Cos θ =1
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Cosec θ . Sin θ =1
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Cot θ . Tan θ =1
Values of Trigonometric Ratios
|
Angles |
[0^{circ} ] |
[30^{circ} ] |
[45^{circ} ] |
[60^{circ} ] |
[90^{circ} ] |
|
Angle (In Radian) |
0 |
[frac{pi}{6}] |
[frac{pi}{4}] |
[frac{pi}{3}] |
[frac{pi}{2}] |
|
Sin θ |
0 |
[ frac{1}{2}] |
[ frac{1}{sqrt{2}}] |
[ frac{sqrt{3}}{2}] |
1 |
|
Cos θ |
1 |
[ frac{sqrt{3}}{2}] |
[ frac{1}{sqrt{2}}] |
[ frac{1}{2}] |
0 |
|
Tan θ |
0 |
[ frac{1}{sqrt{3}}] |
1 |
[sqrt{3}] |
[infty] |
|
Cot θ |
[infty] |
[sqrt{3}] |
1 |
[frac{1}{sqrt{3}} ] |
0 |
|
Sec θ |
1 |
[frac{2}{sqrt{3}} ] |
[sqrt{2}] |
2 |
[infty] |
|
Cosec θ |
[infty] |
2 |
[sqrt{2}] |
[frac{2}{sqrt{3}} ] |
1 |
Sign of Trigonometric Functions
uploaded soon)
Trigonometry Ratios Formula
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Tan θ = sin θ/cos θ
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Cot θ = cos θ/sin θ
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Sin θ = tan θ/cos θ
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Cos θ = sin θ/tan θ
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Sec θ = tan θ/sin θ
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Cosec θ = cos θ/tan θ
Also,
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sin (90°- θ) = cos θ
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cos (90°- θ) = sin θ
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tan (90°- θ) = cot θ
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cot (90°- θ) = tan θ
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sec (90°- θ) = cosec θ
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cosec (90°- θ) = sec θ
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sin (90°+ θ) = cos θ
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cos (90°+ θ) = -sin θ
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tan (90°+ θ) = -cot θ
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cot (90°+ θ) = -tan θ
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sec (90°+ θ) = -cosec θ
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cosec (90°+ θ) = sec θ
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sin (180°- θ) = sin θ
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cos (180°- θ) = -cos θ
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tan (180°- θ) = -tan θ
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cot (180°- θ) = -cot θ
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sec (180°- θ) = -sec θ
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cosec (180°- θ) = cosec θ
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sin (180°+ θ) = -sin θ
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cos (180°+ θ) = -cos θ
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tan (180°+ θ) = tan θ
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cot (180°+ θ) = cot θ
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sec (180°+ θ) = -sec θ
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cosec (180°+ θ) = -cosec θ
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sin (360°- θ) = -sin θ
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cos (360°- θ) = cos θ
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tan (360°- θ) = -tan θ
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cot (360°- θ) = -cot θ
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sec (360°- θ) = sec θ
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cos
ec (360°- θ) = -cosec θ
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sin (360°+ θ) = sin θ
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cos (360°+ θ) = cos θ
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tan (360°+ θ) = -tan θ
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cot (360°+ θ) = -cot θ
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sec (360°+ θ) = sec θ
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cosec (360°+ θ) = -cosec θ
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sin (270°- θ) = -cos θ
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cos (270°- θ) = -sin θ
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sin (270°+ θ) = -cos θ
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cos (270°+ θ) = sin θ
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Tan θ = sin θ/cos θ
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Cot θ = cos θ/sin θ
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Sin θ = tan θ/cos θ
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Cos θ = sin θ/tan θ
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Sec θ = tan θ/sin θ
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Cosec θ = cos θ/tan θ
From that we can say that,
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Sec θ . Cos θ =1
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Cosec θ . Sin θ =1
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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
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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.
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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°.