Trig Review Worksheet - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. We will use trigonometric ratios (Sine, Cosine, Tangent) and the Pythagorean theorem where needed.

Remember SOH CAH TOA:
* Sine = Opposite / Hypotenuse
* Cosine = Adjacent / Hypotenuse
* Tangent = Opposite / Adjacent

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Part 1: Find the measure of each angle indicated (Round to nearest tenth)

1) Find angle $\theta$ at A
* Identify sides relative to angle $\theta$: Adjacent = 12, Hypotenuse = 13.
* Use Cosine: $\cos(\theta) = \frac{12}{13}$
* Calculate inverse cosine: $\theta = \cos^{-1}(\frac{12}{13}) \approx 22.619...^\circ$
* Round to nearest tenth: 22.6°

2) Find angle $\theta$ at A
* Identify sides relative to angle $\theta$: Opposite = 4, Hypotenuse = 13.
* Use Sine: $\sin(\theta) = \frac{4}{13}$
* Calculate inverse sine: $\theta = \sin^{-1}(\frac{4}{13}) \approx 17.92...^\circ$
* Round to nearest tenth: 17.9°

3) Find angle $\theta$ at A
* Identify sides relative to angle $\theta$: Adjacent = 6, Hypotenuse = 9.
* Use Cosine: $\cos(\theta) = \frac{6}{9}$
* Calculate inverse cosine: $\theta = \cos^{-1}(\frac{6}{9}) \approx 48.189...^\circ$
* Round to nearest tenth: 48.2°

4) Find angle $\theta$ at B
* Identify sides relative to angle $\theta$: Adjacent = 10, Hypotenuse = 11.8.
* Use Cosine: $\cos(\theta) = \frac{10}{11.8}$
* Calculate inverse cosine: $\theta = \cos^{-1}(\frac{10}{11.8}) \approx 31.89...^\circ$
* Round to nearest tenth: 31.9°

5) Find angle $\theta$ at B
* Identify sides relative to angle $\theta$: Opposite = 7.7, Hypotenuse = 14.
* Use Sine: $\sin(\theta) = \frac{7.7}{14}$
* Calculate inverse sine: $\theta = \sin^{-1}(0.55) \approx 33.367...^\circ$
* Round to nearest tenth: 33.4°

6) Find angle $\theta$ at B
* Identify sides relative to angle $\theta$: Adjacent = 4, Hypotenuse = 5.
* Use Cosine: $\cos(\theta) = \frac{4}{5}$
* Calculate inverse cosine: $\theta = \cos^{-1}(0.8) \approx 36.869...^\circ$
* Round to nearest tenth: 36.9°

7) Find angle $\theta$ at B
* Identify sides relative to angle $\theta$: Opposite = 11, Adjacent = 4.4.
* Use Tangent: $\tan(\theta) = \frac{11}{4.4}$
* Calculate inverse tangent: $\theta = \tan^{-1}(2.5) \approx 68.198...^\circ$
* Round to nearest tenth: 68.2°

8) Find angle $\theta$ at B
* Identify sides relative to angle $\theta$: Opposite = 3, Adjacent = 3.
* Use Tangent: $\tan(\theta) = \frac{3}{3} = 1$
* Calculate inverse tangent: $\theta = \tan^{-1}(1) = 45^\circ$
* Answer: 45.0°

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Part 2: Find the measure of each side indicated (Round to nearest tenth)

9) Find side $x$ (Hypotenuse)
* Given: Angle = 37°, Adjacent side = 11.
* Use Cosine: $\cos(37^\circ) = \frac{11}{x}$
* Rearrange to solve for $x$: $x = \frac{11}{\cos(37^\circ)}$
* Calculate: $x \approx \frac{11}{0.7986} \approx 13.77$
* Round to nearest tenth: 13.8

10) Find side $x$ (Opposite)
* Given: Angle = 36°, Adjacent side = 13.
* Use Tangent: $\tan(36^\circ) = \frac{x}{13}$
* Rearrange to solve for $x$: $x = 13 \cdot \tan(36^\circ)$
* Calculate: $x \approx 13 \cdot 0.7265 \approx 9.44$
* Round to nearest tenth: 9.4

11) Find side $x$ (Adjacent)
* Given: Angle = 50.1°, Opposite side = 5.
* Use Tangent: $\tan(50.1^\circ) = \frac{5}{x}$
* Rearrange to solve for $x$: $x = \frac{5}{\tan(50.1^\circ)}$
* Calculate: $x \approx \frac{5}{1.196} \approx 4.18$
* Round to nearest tenth: 4.2

12) Find side $x$ (Opposite)
* Given: Angle = 60°, Hypotenuse = 11.
* Use Sine: $\sin(60^\circ) = \frac{x}{11}$
* Rearrange to solve for $x$: $x = 11 \cdot \sin(60^\circ)$
* Calculate: $x \approx 11 \cdot 0.866 \approx 9.526$
* Round to nearest tenth: 9.5

Final Answer:
1) 22.6°
2) 17.9°
3) 48.2°
4) 31.9°
5) 33.4°
6) 36.9°
7) 68.2°
8) 45.0°
9) 13.8
10) 9.4
11) 4.2
12) 9.5