The pythagorean trigonometric identity can be used to compute cos(θ) when sin(θ) is known. The formula is sin²(θ) + cos²(θ) = 1. Thus, cos(θ) is computed by taking the square root of (1 - sin²(θ)).
Continue ReadingRaise sin(θ) to the power 2. For any real value of the angle θ between −∞ and ∞, the function sin(θ) can only take values from the interval [-1, 1]. For this reason, sin²(θ) is always found between zero and one. Assuming that θ = 30 degrees, sin(θ) will be 0.5 and sin²(θ) is 0.25.
Subtract the value of sin²(θ) from 1. Since the co-domain of sin²(θ) is the interval [0, 1], it follows that the function 1 - sin²(θ) is also restricted to values between zero and one. For θ = 30 degrees, 1 - sin²(θ) is 0.75 or 3/4.
Compute the square root of (1 - sin²(θ)). Since this function takes positive values between zero and one, the square root will also have real values. If θ = 30 degrees, √(1 - sin²(θ)) will be √(3/4) or √3/2. This is precisely the value of cos(θ) for θ = 30 degrees.