I want to solve the following equation using trigonometric identities without the use of a calculator. In order to find the solution, I used the double angle trigonometric identity.Unfortunately, it seems I have made a mistake in my calculation and I cannot seem to find it.
I want to find $sin(\dfrac{-7\pi}{12})$.
I know from the double angle trigonometric identity that
$$sin\left(\dfrac{\left(\dfrac{-7\pi}{6}\right)}{2}\right)=\sqrt{\dfrac{1-cos\left(\dfrac{-7\pi}{6}\right)}{2}}$$
$$sin(\dfrac{-7\pi}{12})=\sqrt{\dfrac{1–\dfrac{\sqrt{3}}{2}}{2}}$$
$$sin(\dfrac{-7\pi}{12})=\sqrt{\dfrac{2+\sqrt{3}}{4}}$$
$$sin(\dfrac{-7\pi}{12})=\dfrac{\sqrt{2+\sqrt{3}}}{2}$$
Looking online for a simplifier I get a different answer… What went wrong?
Best Answer
Watch for your signs when using $$sin\left(\dfrac{\left(\dfrac{-7\pi}{6}\right)}{2}\right)= \pm \sqrt{\dfrac{1-cos\left(\dfrac{-7\pi}{6}\right)}{2}}$$
You may have a negative solution as well.