The double-angle identity for sine is derived using the sum identity sin(𝐴+𝐵)=sin(𝐴)cos(𝐵)+cos(𝐴)sin(𝐵). When 𝐴=𝐵=𝜃, it simplifies to 2sin(𝜃)cos(𝜃)
The inverse cosine function, cos−1(𝑥)cos −1 (x), gives the angle whose cosine is 𝑥. When cos(𝜃)=0, the angle θ is 𝜋/2 (or 90°) in the range of [0,𝜋].
This is a fundamental Pythagorean identity in trigonometry:
1+tan2(θ) = sec 2(θ).
It is derived from dividing the basic identity
sin 2(𝜃)+cos2(𝜃)=1 by cos2(𝜃).
This is one of the double-angle identities for cosine:
cos(2θ)=cos2(θ)−sin 2(θ).
Using 2sin(𝑥)cos(𝑥)=sin(2𝑥)the equation becomes sin(2x) = √3/2. The solutions for 2𝑥=𝜋/3 and 2x = 4π/3 yield 𝑥=𝜋/3 and x = 4π/3 within the given interval.
