Trigonometric Identities: Precalculus Math Test – PDF Download

Precalculus Trig Identities Test: questions 1–5

Question 1: Six functions from a point (3 points)

The terminal side of angle \(\theta\) passes through the point shown.

1. Find \(\sin\theta\) and \(\cos\theta\). (1 point)
2. Find \(\tan\theta\) and \(\text{sec}\,\theta\). (1 point)
3. Show that the values from part (a) satisfy \(\sin^{2}\theta+\cos^{2}\theta=1\). (1 point)

Question 2: Values from a Pythagorean identity (2 points)

Angle \(\theta\) is in Quadrant IV and \(\sin\theta=-\frac{9}{41}\).

1. Find \(\cos\theta\). (1 point)
2. Find \(\tan\theta\). (1 point)

Question 3: Simplify (2 points)

Simplify each expression to a single term.

1. \((\text{sec}^{2}\theta-1)\cos^{2}\theta\) (1 point)
2. \(\frac{\sin\theta}{1-\cos^{2}\theta}\) (1 point)

Question 4: Verify an identity (3 points)

Verify the identity \(\frac{1+\sin\theta}{\cos\theta}=\frac{\cos\theta}{1-\sin\theta}\).

Question 5: Exact value of sin 105° (2 points)

Find the exact value of \(\sin 105^{\circ}\) and explain which angles you used.

Precalculus Trig Identities Test: questions 6–8

Question 6: Difference formulas (3 points)

Angle \(A\) is in Quadrant II with \(\sin A=\frac{8}{17}\). Angle \(B\) is in Quadrant IV with \(\cos B=\frac{3}{5}\).

1. Find \(\cos A\) and \(\sin B\). (1 point)
2. Find \(\cos(A-B)\). (1 point)
3. Find \(\sin(A-B)\). (1 point)

Question 7: Double angle from a tangent (3 points)

Angle \(\theta\) is in Quadrant III and \(\tan\theta=\frac{4}{3}\).

1. Find \(\sin\theta\) and \(\cos\theta\). (1 point)
2. Find \(\sin 2\theta\) and \(\cos 2\theta\). (1 point)
3. Find \(\tan 2\theta\) two different ways. (1 point)

Question 8: Prove a double-angle identity (2 points)

Prove that \(\cos 2\theta=\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}\).

How to use this precalculus trig identities test

Take this precalculus trig identities test without notes, then grade it with the test answer key out of 20 points.

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