Trigonometric Identities: Precalculus Math Quiz – PDF Download
Ten multiple-choice questions on trigonometric identities: Pythagorean and reciprocal identities, sum and difference formulas, and double angles. Choose the single best answer, then read the explanation to learn the typical mistake.
PrecalculusCommon Core: F-TF.C.8 · F-TF.C.9
🎯 Trigonometric Identities – Precalculus math quiz
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Question 1
Point P is on the unit circle in Quadrant II and its x-coordinate is \(-\frac{5}{13}\). What is \(\sin\theta\)?Point P on the unit circle.
A\(-\frac{12}{13}\)
B\(\frac{12}{13}\)
C\(\frac{5}{13}\)
D\(\frac{8}{13}\)
Question 2
Simplify \(\sin\theta\,\text{cot}\,\theta\).
A\(\text{sec}\,\theta\)
B\(\tan\theta\)
C\(\cos\theta\)
D\(1\)
Question 3
Which equation is true for every angle where both sides are defined?
A\(1+\tan\theta=\text{sec}\,\theta\)
B\(\sin\theta+\cos\theta=1\)
C\(1+\tan^{2}\theta=\text{sec}^{2}\theta\)
D\(\sin^{2}\theta-\cos^{2}\theta=1\)
Question 4
If \(\sin\theta=\frac{3}{5}\), what is \(\cos 2\theta\)?
A\(\frac{24}{25}\)
B\(-\frac{1}{5}\)
C\(\frac{9}{25}\)
D\(\frac{7}{25}\)
Question 5
The figure shows rays at 45 and 75 degrees. Which is the exact value of \(\sin 75^{\circ}\)?75 degrees is 45 plus 30.
A\(\frac{\sqrt{6}-\sqrt{2}}{4}\)
B\(\frac{\sqrt{2}+1}{2}\)
C\(\frac{\sqrt{2}}{4}\)
D\(\frac{\sqrt{6}+\sqrt{2}}{4}\)
Question 6
If \(\tan A=\frac{1}{2}\) and \(\tan B=\frac{1}{3}\), what is \(\tan(A+B)\)?
A\(1\)
B\(\frac{5}{6}\)
C\(\frac{5}{7}\)
D\(\frac{1}{6}\)
Question 7
Which expression is equal to \(\frac{\sin\theta}{1-\cos^{2}\theta}\)?
A\(\sin\theta\)
B\(\text{csc}\,\theta\)
C\(\text{sec}\,\theta\)
D\(\tan\theta\)
Question 8
The graph shows \(y=2\sin x\cos x\). Which function has the same graph?A sine wave with period pi.
A\(y=\sin 2x\)
B\(y=2\sin x\)
C\(y=\cos 2x\)
D\(y=\sin\frac{x}{2}\)
Question 9
Angle \(\theta\) is in Quadrant III and \(\cos\theta=-\frac{4}{5}\). What is \(\tan\theta\)?
A\(-\frac{3}{4}\)
B\(\frac{4}{3}\)
C\(-\frac{4}{3}\)
D\(\frac{3}{4}\)
Question 10
Which expression is equal to \(\sin(A-B)\)?
A\(\sin A\cos B-\cos A\sin B\)
B\(\sin A\cos B+\cos A\sin B\)
C\(\cos A\cos B-\sin A\sin B\)
D\(\sin A-\sin B\)
🎉 Quiz complete!
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Remember:
Standards F-TF.C.8 and F-TF.C.9: prove and use the Pythagorean identity, and the addition formulas for sine, cosine and tangent.
Pythagorean identities: \(\sin^{2}\theta+\cos^{2}\theta=1\), \(1+\tan^{2}\theta=\text{sec}^{2}\theta\) and \(1+\text{cot}^{2}\theta=\text{csc}^{2}\theta\).
Sum formulas: \(\sin(A+B)=\sin A\cos B+\cos A\sin B\) and \(\cos(A+B)=\cos A\cos B-\sin A\sin B\).
Double angle: \(\sin 2\theta=2\sin\theta\cos\theta\) and \(\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta\).
To verify an identity, work on one side only, rewrite in sine and cosine, and simplify until it matches the other side.
