What are trigonometric identities?

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. The trigonometric identities hold only for the right-angle triangle.

Trigonometric identities are useful whenever trigonometric functions are involved in an expression or an equation. Trigonometric identities are true for every value of variables being on both sides of an equation. Geometrically, these identities involve certain trigonometric functions ( similar to sine, cosine, tangent) of one or further angles.

Primary trigonometric identities

Sine, cosine, and tangent are the primary trigonometric functions, whereas cotangent, secant, and cosecant are the other three functions. The trigonometric identities are based on all six trig functions. All the trigonometric identities are grounded on the six trigonometric ratios. They're sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right-angled triangle, a conterminous side, a contrary side, and a hypotenuse side. All the abecedarian trigonometric identities are deduced from the six trigonometric rules.

There are many identities in trigonometry which are used to break numerous trigonometric problems. Using these trigonometric identities or formulas, complex trigonometric questions can be answered snappily. Let us see all the trigonometric identities.

Reciprocal trigonometric identities

The reciprocal trigonometric identities are:

• $\sin \theta = \frac{1}{\cos ec\theta }$ or $\cos ec\theta = \frac{1}{\sin \theta }$
• $\cos \theta = \frac{1}{sec\theta }$ or $sec\theta = \frac{1}{\cos \theta }$
• $\tan \theta = \frac{1}{\mathrm{co}t\theta }$ or $cot\theta = \frac{1}{\tan \theta }$

Pythagorean trigonometric identities

There are three Pythagorean trigonometric identities in trigonometry that are based on the right-angled triangle theorem or Pythagoras theorem. The identities are:

• ${\sin }^2\theta + {\cos }^2\theta = 1$
• $1 + {\tan }^2\theta = se{c}^2\theta$
• $1 + co{t}^2\theta =\cos e{c}^2\theta$

Ratios of trigonometric identities

The ratios of trigonometric identities are:

• $\tan \theta = \frac{\sin \theta }{\cos \theta }$
• $cot\theta = \frac{\cos \theta }{\sin \theta }$

Trigonometric identities of opposite angles

The list of opposite angle trigonometric identities are:

$\sin (-\theta ) = -\sin \theta$

$\cos (-\theta ) = \cos \theta$

$\tan (-\theta ) = -\tan \theta$

$cot(-\theta ) = -cot\theta$

$sec(-\theta ) = sec\theta$

$\cos ec(-\theta ) = -\cos ec\theta$

Trigonometric identities of complementary angles

In geometry, two angles are complementary if their sum is equal to 90 degrees. Similarly, we can learn here the trigonometric identities for complementary angles.

• $\sin ({90}^{\circ }-\theta ) = \cos \theta$
• $\cos ({90}^{\circ }-\theta ) = \sin \theta$
• $\tan ({90}^{\circ }-\theta ) = \mathrm{co}t\theta$
• $cot({90}^{\circ }-\theta ) = \tan \theta$
• $sec({90}^{\circ }-\theta ) = \cos ec\theta$
• $\cos ec({90}^{\circ }-\theta ) = sec\theta$

Trigonometric identities of supplementary angles

Two angles are supplementary if their sum is equal to 90 degrees. Similarly, we can learn here the trigonometric identities for supplementary angles.

• $\sin ({180}^{\circ }-\theta ) = \sin \theta$
• $\cos ({180}^{\circ }-\theta ) = -\cos \theta$
• $\cos ec({180}^{\circ }-\theta ) = \cos ec\theta$
• $sec({180}^{\circ }-\theta ) = -sec\theta$
• $\tan ({180}^{\circ }-\theta ) = -\tan \theta$
• $\mathrm{co}t({180}^{\circ }-\theta ) = -\mathrm{co}t\theta$

Sum and difference of angles of trigonometric identities

Consider two angles, $\alpha$ and $\beta$. The trigonometric sum and difference identities are as follows:

• $\sin (\alpha + \beta ) = \sin \alpha .\cos \beta + \cos \alpha .\sin \beta$
• $\sin (\alpha - \beta ) = \sin \alpha .\cos \beta - \cos \alpha .\sin \beta$
• $\cos (\alpha + \beta ) = \cos \alpha .\cos \beta - \sin \alpha .\sin \beta$
• $\cos (\alpha - \beta ) = \cos \alpha .\cos \beta + \sin \alpha .\sin \beta$
• $\tan (\alpha +\beta ) = \frac{\tan \alpha +\tan \beta }{1-\tan \alpha .\tan \beta }$
• $\tan (\alpha -\beta ) = \frac{\tan \alpha -\tan \beta }{1+\tan \alpha .\tan \beta }$

Double angle trigonometric identities

If the angles are doubled, then the trigonometric identities for sin, cos, and tan are:

• $\sin 2\theta = 2\sin \theta .\cos \theta$
• $\cos 2\theta = {\cos }^2\theta - {\sin }^2\theta = 2{\cos }^2\theta - 1 = 1 - 2{\sin }^2\theta$
• $\tan 2\theta = \frac{2\tan \theta }{1-{\tan }^2\theta }$

Half angle identities

If the angles are halved, then the trigonometric identities for sin, cos, and tan are:

• $\sin \frac{\theta }{2} = \pm \sqrt{\frac{1-\cos \theta }{2}}$
• $\cos \frac{\theta }{2}=\pm \sqrt{\frac{1+\cos \theta }{2}}$
• $\tan \frac{\theta }{2}=\pm \frac{\sin \theta }{1+\cos \theta }$

Product-Sum trigonometric identities

The product-sum trigonometric identities change the sum or difference of sines or cosines into a product of sines and cosines.

• $\sin A+\sin B = 2.\sin \left(\frac{A+B}{2}\right).\cos \left(\frac{A-B}{2}\right)$
• $\cos A+\cos B = 2.\cos \left(\frac{A+B}{2}\right).\cos \left(\frac{A-B}{2}\right)$
• $\sin A-\sin B = 2.\cos \left(\frac{A+B}{2}\right).\sin \left(\frac{A-B}{2}\right)$
• $\cos A-\cos B = -2.\sin \left(\frac{A+B}{2}\right).\sin \left(\frac{A-B}{2}\right)$

Trigonometric identities of products

These identities are:

• $\sin A.\sin B = \frac{1}{2}\left[\cos \right(A-B) - \cos (A+B\left)\right]$
• $\sin A.\cos B = \frac{1}{2}\left[\sin \right(A+B)+\sin (A-B\left)\right]$
• $\cos A.\cos B =\frac{1}{2} \left[\cos \right(A+B)+\cos (A-B\left)\right]$
• $\cos A.\sin B =\frac{1}{2}\left[\sin (A+B) -\sin (A-B)\right]$

Triangle identities (Sine, Cosine, Tangent rule)

They are the triangle identities. The identities or equations are applicable for all the triangles and not just for the right triangles. These identities include:

• Sine law
• Cosine law
• Tangent law

If A, B, and C are the vertices of a triangle and a, b and c are the respective sides, then according to the sine law or sine rule,

$\frac{a}{\sin A} = \frac{b}{\sin B} =\frac{c}{\sin C}$

or

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

According to cosine law,

$c^2 = a^2+b^2-2ab.\cos C$

or

$\cos C = \frac{a^2+b^2-c^2}{2ab}$

According to tangent law,

$\frac{a-b}{a+b} = \frac{\tan \left({\displaystyle \frac{A-B}{2}}\right)}{\tan \left({\displaystyle \frac{A+B}{2}}\right)}$

Common Mistakes

Students generally get confused and forget about different trigonometric identities. They face difficulties in remembering ‘+’ and ‘-’ signs in different identities.

Context and Applications

The trigonometric identities are useful in measuring the height of buildings, mountains, and other dimensions of objects that form a right-angled triangle. Trigonometry is used on daily basis in aviation, marine, navigation, and so on. The subject is important in high school, senior secondary schools, and graduate level, and all the exams relating to mathematics, engineering, or science field, particularly for Bachelors of Mathematics, Bachelors of Science, Bachelors of Engineering and Technology, and Masters in Engineering and Technology.

Practice Problems

1. Which of the following is the equality that involves trigonometry functions and holds for all the values of variables given in the equation?

1. Trigonometric identities
2. Ratios and Proportions
3. Multiplicative inverse
4. Factorization

Answer – a

Explanation – Trigonometric identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation. A ratio is a relationship between two properties and proportion is an equation in which ratios are set equal. The multiplicative inverse of a number is the one that when multiplied by its original number, gives an answer equal to 1. Factorization is the process of deriving factors of a certain number.

2. Which of the following is not a trigonometric identity?

1. Tangent
2. Sine
3. Reynolds
4. Cosine

Answer – c

Explanation – Reynolds is not a trigonometric identity. Tangent, sine, and cosine are all the trigonometric identities.

3. Which of the following is not a primary trigonometric function?

1. sine
2. cosine
3. tangent
4. secant

Answer – d

Explanation – The secant is not a primary trigonometric function. Sine, cosine, and tangent are the primary trigonometric functions.

4. Which of the triangles is used to define the trigonometric ratios?

1. equilateral triangle
2. isosceles triangle
3. scalene triangle
4. right-angled triangle

Answer – d

Explanation – A right-angled triangle is used to define the trigonometric ratios. The equilateral triangles, isosceles triangles, and scalene triangles are not used for defining the trigonometric ratios.

5. Which are the two angles complementary in geometry?

1. if their sum is equal to 45 degrees
2. if their sum is equal to 90 degrees
3. if their sum is equal to 180 degrees
4. if their sum is equal to 270 degrees

Answer – b

Explanation – In geometry, two angles are complementary if their sum is equal to 90 degrees.