How Do You Get Cos Theta From Sin Theta?

How Do You Get Cos Theta From Sin Theta?

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)).

  1. Compute sin²(θ)

    Raise 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.

  2. Compute 1 - sin²(θ)

    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.

  3. Take the square root of (1 - sin²(θ))

    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.