How to use related angle identities in trigonometry

Key takeaway: These identities come up constantly in trig simplification. The main job is spotting whether the angle is $\pi-x$, $\pi+x$, $2\pi-x$, or $-x$, then tracking the sign correctly.

Here are the related-angle identities in a cleaner form:

• $\sin(\pi-x)=\sin x$

• $\cos(\pi-x)=-\cos x$

• $\tan(\pi-x)=-\tan x$

• $\sin(\pi+x)=-\sin x$

• $\cos(\pi+x)=-\cos x$

• $\tan(\pi+x)=\tan x$

• $\sin(2\pi-x)=-\sin x$

• $\cos(2\pi-x)=\cos x$

• $\tan(2\pi-x)=-\tan x$

• $\sin(-x)=-\sin x$

• $\cos(-x)=\cos x$

• $\tan(-x)=-\tan x$

A quick way to remember them is to think about the unit circle:

• sine changes sign when the point moves above/below the $x$-axis,
• cosine changes sign when the point moves left/right of the $y$-axis,
• tangent follows the ratio $\sin x / \cos x$.

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Pitfalls the pros know 👇 The usual mistake is mixing up $\pi-x$ and $\pi+x$. They do not keep the same signs. Another one is forgetting that cosine is even, so $\cos(-x)=\cos x$, while sine and tangent are odd.

What if the problem changes? If the angle is written as $\frac{3\pi}{2}-x$ or $x-\pi$, convert it first into one of the standard forms. That saves time and prevents sign errors. For example, $\sin(x-\pi)=-\sin x$ because it matches the $\pi+x$ pattern after rearranging.

Tags: unit circle, even and odd functions, reference angle