Summer brush-up · Math

Trigonometry check: ready for Precalculus / Calculus or Statistics?

For students who finished Trigonometry (usually grades 11–12) or are deciding whether to start Precalculus / Calculus or Statistics.

14 items drawn from the Trigonometry final exam. About 30–40 minutes. Paper and pencil, no calculator unless an item says so. Work every item first, then check each answer and mark it.

Print test with answer key

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1. Solve right triangle ABC with C = 90^\circ, A = 34^\circ and hypotenuse c = 50. Round sides to the nearest tenth.

Answer: B = 56^\circ, a = 50\sin 34^\circ \approx 28.0, b = 50\cos 34^\circ \approx 41.5
Lesson 1.4 Finding Unknown Angles

2. If \cos\theta = \dfrac{2}{7} and \theta is acute, find the exact values of \csc\theta and \cot\theta with rationalized denominators.

Answer: The opposite side is \sqrt{49 - 4} = 3\sqrt{5}, so \csc\theta = \dfrac{7}{3\sqrt{5}} = \dfrac{7\sqrt{5}}{15} \approx 1.0435 and \cot\theta = \dfrac{2}{3\sqrt{5}} = \dfrac{2\sqrt{5}}{15} \approx 0.2981
Lesson 1.5 The Reciprocal Ratios

3. Find the exact value of \tan 30^\circ \cdot \sec 30^\circ.

Answer: \dfrac{1}{\sqrt{3}}\cdot\dfrac{2}{\sqrt{3}} = \dfrac{2}{3}
Lesson 2.2 The 30-60-90 Triangle

4. The minute hand of the clock in a church tower is 15 cm long. How far does its tip travel in 20 minutes? Give the exact answer and round to the nearest tenth of a centimeter.

Answer: \theta = \dfrac{20}{60}\cdot 2\pi = \dfrac{2\pi}{3}, so s = 15\cdot\dfrac{2\pi}{3} = 10\pi \approx 31.4 cm
Lesson 3.4 Arc Length and Sector Area

5. At a sawmill, a belt connects a pulley of radius 4 in to a pulley of radius 10 in. The small pulley turns at 300 revolutions per minute. How fast does the large pulley turn, in revolutions per minute?

Answer: Both rims move at the belt's linear speed, so 4\cdot 300 = 10\,\omega and \omega = 120 rpm
Lesson 3.5 Linear and Angular Speed

6. For y = 2\cos(3x + \pi) + 1, give the amplitude, period, phase shift, midline, maximum and minimum. (Hint: factor the inside first.)

Answer: y = 2\cos\left(3\left(x + \dfrac{\pi}{3}\right)\right) + 1: amplitude 2, period \dfrac{2\pi}{3}, phase shift \dfrac{\pi}{3} to the left, midline y = 1, maximum 3, minimum -1
Lesson 4.4 Phase Shift and Vertical Shift

7. For y = \tan(2x), find the period and the vertical asymptotes in [0, \pi), exact in radians.

Answer: Period \dfrac{\pi}{2}; asymptotes x = \dfrac{\pi}{4} and x = \dfrac{3\pi}{4}
Lesson 4.6 The Graph of Tangent

8. Simplify \dfrac{1 - \cos^2 x}{\sin x\cos x} to a single trigonometric function.

Answer: \dfrac{\sin^2 x}{\sin x\cos x} = \dfrac{\sin x}{\cos x} = \tan x
Lesson 5.2 The Pythagorean Identities

9. Use a sum formula to find the exact value of \sin 75^\circ.

Answer: \sin(45^\circ + 30^\circ) = \dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659
Lesson 5.4 Sum and Difference Formulas

10. Find the exact values in radians: (a) \arccos\left(-\dfrac{1}{2}\right) (b) \arctan\left(-\sqrt{3}\right)

Answer: (a) \dfrac{2\pi}{3} (b) -\dfrac{\pi}{3}
Lesson 6.2 Inverse Cosine and Inverse Tangent

11. Solve 2\cos x + \sqrt{3} = 0 on [0, 2\pi). Give exact answers in radians.

Answer: \cos x = -\dfrac{\sqrt{3}}{2}, so x = \dfrac{5\pi}{6}, \dfrac{7\pi}{6}
Lesson 6.4 Solving Basic Trigonometric Equations

12. A triangular field on Mr. Miller's farm has two fences of 120 m and 150 m that meet at an angle of 48^\circ. Find the area of the field to the nearest square meter.

Answer: \dfrac{1}{2}(120)(150)\sin 48^\circ \approx 6688 m^2
Lesson 7.1 The Area of a Triangle

13. Challenge. In triangle ABC, A = 35^\circ, a = 8 and b = 12. How many triangles fit these data? Find every possible measure of angle B, to the nearest tenth of a degree.

Answer: \sin B = \dfrac{12\sin 35^\circ}{8} \approx 0.8604. Both B \approx 59.4^\circ and B \approx 120.6^\circ work, since 35^\circ + 120.6^\circ < 180^\circ: two triangles
Lesson 7.3 The Ambiguous Case (SSA)

14. From a point on level ground 60 ft from the base of a grain silo, the angle of elevation to the top of the silo is 52^\circ. Find the height of the silo to the nearest tenth of a foot.

Answer: 60\tan 52^\circ \approx 76.8 ft
Lesson 8.1 Angles of Elevation and Depression

4-week summer review path

Four short days a week, 30–45 minutes: the 16 Trigonometry lessons that appear most often on its exams and reviews, in course order. Do the lessons you missed on the test first.

Trigonometry check: ready for Precalculus / Calculus or Statistics?

4-week summer review path

Week 1

Week 2

Week 3

Week 4