Trigonometric identities are equations that relate to different trigonometric functions and are true for any value of the variable that is there in the domain. Basically, an identity is an equation that holds true for all the values of the variable(s) present in it.
For example, some of the algebraic identities are:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab+ b2
(a + b)(a-b)= a2 – b2
The algebraic identities relate just the variables whereas the trig identities relate the 6 trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. Let’s learn about each type of trigonometric identities in detail.
Reciprocal Trigonometric Identities
We already know that the reciprocals of sin, cosine, and tangent are cosecant, secant, and cotangent respectively.
Thus, the reciprocal identities are given as,
- sin θ = 1/cosecθ (OR) cosec θ = 1/sinθ
- cos θ = 1/secθ (OR) sec θ = 1/cosθ
- tan θ = 1/cotθ (OR) cot θ = 1/tanθ
Pythagorean Trigonometric Identities
The Pythagorean trigonometric identities in trigonometry are derived from the Pythagoras theorem. Applying Pythagoras theorem to the right-angled triangle below, we get:
Opposite2 + Adjacent2 = Hypotenuse2
Dividing both sides by Hypotenuse2
Opposite2/Hypotenuse2 + Adjacent2/Hypotenuse2 = Hypotenuse2/Hypotenuse2
- sin2θ + cos2θ = 1
This is one of the Pythagorean identities. In the same way, we can derive two other Pythagorean trigonometric identities.
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ
Complementary and Supplementary Trigonometric Identities
The complementary angles are a pair of two angles such that their sum is equal to 90°. The complement of an angle θ is (90 – θ). The trigonometric ratios of complementary angles are:
- sin (90°- θ) = cos θ
- cos (90°- θ) = sin θ
- cosec (90°- θ) = sec θ
- sec (90°- θ) = cosec θ
- tan (90°- θ) = cot θ
- cot (90°- θ) = tan θ
The supplementary angles are a pair of two angles such that their sum is equal to 180°. The supplement of an angle θ is (180 – θ). The trigonometric ratios of supplementary angles are:
- sin (180°- θ) = sinθ
- cos (180°- θ) = -cos θ
- cosec (180°- θ) = cosec θ
- sec (180°- θ)= -sec θ
- tan (180°- θ) = -tan θ
- cot (180°- θ) = -cot θ
Sum and Difference Trigonometric Identities
The sum and difference identities include the formulas of sin(A+B), cos(A-B), cot(A+B), etc.
- sin (A+B) = sin A cos B + cos A sin B
- sin (A-B) = sin A cos B – cos A sin B
- cos (A+B) = cos A cos B – sin A sin B
- cos (A-B) = cos A cos B + sin A sin B
- tan (A+B) = (tan A + tan B)/(1 – tan A tan B)
- tan (A-B) = (tan A – tan B)/(1 + tan A tan B)
Double and Half Angles Trigonometric Identities
Double angle formulas: The double angle trigonometric identities can be obtained by using the sum and difference formulas.
For example, from the above formulas:
sin (A+B) = sin A cos B + cos A sin B
Substitute A = B = θ on both sides here, we get:
sin (θ + θ) = sinθ cosθ + cosθ sinθ
sin 2θ = 2 sinθ cosθ
In the same way, we can derive the other double-angle identities.
- sin 2θ = 2 sinθ cosθ
- cos 2θ = cos2θ – sin 2θ
= 2 cos2θ – 1
= 1 – sin 2 θ - tan 2θ = (2tanθ)/(1 – tan2θ)
Half Angle Formulas
Using one of the above double angle formulas,
cos 2θ = 1 – 2 sin2θ
2 sin2θ = 1- cos 2θ
sin2θ = (1 – cos2θ)/(2)
sin θ = ±√[(1 – cos 2θ)/2]
Replacing θ by θ/2 on both sides,
sin (θ/2) = ±√[(1 – cos θ)/2]
This is the half-angle formula of sin.
In the same way, we can derive the other half-angle formulas.
sin (θ/2) = ±√[(1 – cosθ)/2]
cos (θ/2) = ±√(1 + cosθ)/2
tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]
The trigonometric identities that we have learned are derived using the right-angled triangles. There are a few other identities that we use in the case of triangles that are not right-angled.
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