5.1. Verifying Trigonometric Identities
One thing that makes trigonometry so useful is that there are numerous trigonometric identities. In this section, we state and derive a few of the most basic ones and verify them for various values of the arguments.
We have already seen several trigonometric identities in the previous chapter. These were mainly identities which show how various trigonometric functions transform under translations and reflections, such as sin(π/2 - θ) = cos θ and tan(-θ) = -tan θ. Here we derive some other quite useful identities of a different form.
One of the most basic trigonometric identites is the following:
- (5.1.1) sin2θ + cos2θ = 1
(The notation sin2θ is shorthand for (sin θ)2, etc.) It is easy to prove this equation. Consider Figure 4.2.1. We see that the point P = (x = cos θ, y = sin θ) lies on the unit circle, so its distance to the origin is 1. But we know from the distance formula (1.3.1) that this distance is also equal to d(O,P) = √x2 + y2 = √cos2θ + sin2θ = 1, whence the result follows.
Two similar trigonometric identities are easily derived from (5.1.1), namely the following:
- (5.1.2a) tan2θ + 1 = sec2θ
- (5.1.2a) cot2θ + 1 = csc2θ
Equation (5.1.2a) followes immediately from (5.1.1) by dividing both sides by cos2θ. Similarly, (5.1.2b) follows from (5.1.1) by dividing both sides by sin2θ.
Example 1: Verify (5.1.1) for θ = π/6.
Solution: We must check that sin2(π/6) + cos2(π/6) = 1. The left side is (√3/2)2 + (1/2)2 = 3/4 + 1/4 = 1, verifying the identity.
Example 2: Verify (5.1.2a) for θ = π/4.
Solution: We must check that tan2(π/4) + 1 = sec2(π/4). The left side is 1 + 1 = 2 and the right side is (√2)2 = 2, verifying the identity.
Example 3: Verify (5.1.2b) for θ = π/3.
Solution: We must check that cot2(π/3) + 1 = csc 2(π/3). The left side is (√3/3)2 + 1 = 1/3 + 1 = 4/3 and the right side is (2√3/3)2 = 12/9 = 4/3, verifying the identity.