Exact value practice with sine, cosine, secant, cotangent, and cosecant

Key takeaway: These expressions are built from standard unit-circle values. Once you know the special angles, each part falls out cleanly.

Evaluate each expression exactly.

a) $\sin \frac{\pi}{3} - \cos \frac{\pi}{6}$

$\sin \frac{\pi}{3}=\frac{\sqrt{3}}{2},\qquad \cos \frac{\pi}{6}=\frac{\sqrt{3}}{2}$

So

$\sin \frac{\pi}{3} - \cos \frac{\pi}{6}=0.$

b) $\sec \frac{\pi}{6} + 2\cot \frac{\pi}{4}$

$\sec \frac{\pi}{6}=\frac{1}{\cos \frac{\pi}{6}}=\frac{1}{\sqrt{3}/2}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3},$

and

$\cot \frac{\pi}{4}=1.$

Thus

$\sec \frac{\pi}{6} + 2\cot \frac{\pi}{4}=\frac{2\sqrt{3}}{3}+2.$

c) $\sin^2 \frac{\pi}{6} + \cos^2 \frac{\pi}{4}$

$\sin \frac{\pi}{6}=\frac12 \Rightarrow \sin^2 \frac{\pi}{6}=\frac14,$

$\cos \frac{\pi}{4}=\frac{\sqrt2}{2} \Rightarrow \cos^2 \frac{\pi}{4}=\frac12.$

So

$\frac14+\frac12=\frac34.$

d) $4\sin \frac{\pi}{4} + \sec^2 \pi$

$4\sin \frac{\pi}{4}=4\cdot \frac{\sqrt2}{2}=2\sqrt2,$

and

$\sec \pi=\frac{1}{\cos \pi}=\frac{1}{-1}=-1 \Rightarrow \sec^2\pi=1.$

Therefore

$4\sin \frac{\pi}{4} + \sec^2 \pi=2\sqrt2+1.$

e) $\sqrt[3]{\csc \frac{\pi}{6}} - \sin^2 \frac{\pi}{2}$

$\csc \frac{\pi}{6}=\frac{1}{\sin \frac{\pi}{6}}=2,$

so

$\sqrt[3]{\csc \frac{\pi}{6}}=\sqrt[3]{2}.$

Also,

$\sin \frac{\pi}{2}=1 \Rightarrow \sin^2 \frac{\pi}{2}=1.$

Thus

$\sqrt[3]{\csc \frac{\pi}{6}} - \sin^2 \frac{\pi}{2}=\sqrt[3]{2}-1.$

Final answers

• a) $0$
• b) $\frac{2\sqrt3}{3}+2$
• c) $\frac34$
• d) $2\sqrt2+1$
• e) $\sqrt[3]{2}-1$

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Pitfalls the pros know 👇 Two places people slip are reciprocal functions and exponents. Remember that $\sec x=1/\cos x$ and $\csc x=1/\sin x$, then square only after the reciprocal is taken. Also, $\sec^2\pi$ means $(\sec \pi)^2$, not $\sec(2\pi)$.

What if the problem changes? If the angles were replaced with nearby special angles, the method would be the same: evaluate the base trig value first, then apply any reciprocal, power, or root. If an angle is not a standard one, you would usually need a calculator approximation instead of an exact radical form.

Tags: unit circle values, reciprocal identities, special angles