Trigonometric equations worksheet for solving for x in degrees, with problems involving sin, cos, and tan functions.

Let's solve each of the trigonometric equations for $ 0^\circ \leq x \leq 360^\circ $, rounding answers to the nearest tenth. We'll go step by step and explain key concepts.

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General Strategy

- Use inverse trig functions: $ \sin^{-1}, \cos^{-1}, \tan^{-1} $
- Consider the quadrants where each function is positive/negative.
- For sine: positive in Q1 & Q2
- For cosine: positive in Q1 & Q4
- For tangent: positive in Q1 & Q3
- Use identities like $ \sin(180^\circ - x) = \sin x $, $ \cos(360^\circ - x) = \cos x $, etc.
- For equations involving $ \sin(nx), \cos(nx), \tan(nx) $, solve for the inner angle first, then solve for $ x $.

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Solutions

#### 1) $ \sin x = 1 $
- $ \sin x = 1 $ at $ x = 90^\circ $
- Only one solution in $ [0^\circ, 360^\circ] $
- ✔ Answer: $ x = 90.0^\circ $

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#### 2) $ \tan x = 1 $
- $ \tan x = 1 $ at $ x = 45^\circ $ and $ x = 225^\circ $ (Q1 and Q3)
- ✔ Answer: $ x = 45.0^\circ, 225.0^\circ $

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#### 3) $ \cos x = 1 $
- $ \cos x = 1 $ at $ x = 0^\circ $ and $ x = 360^\circ $
- Since both are included, list both
- ✔ Answer: $ x = 0.0^\circ, 360.0^\circ $

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#### 4) $ \sin x = 0.5 $
- $ \sin^{-1}(0.5) = 30^\circ $
- Sine is positive in Q1 and Q2
- So $ x = 30^\circ $ and $ 180^\circ - 30^\circ = 150^\circ $
- ✔ Answer: $ x = 30.0^\circ, 150.0^\circ $

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#### 5) $ \cos x = 0.6 $
- $ \cos^{-1}(0.6) \approx 53.1^\circ $
- Cosine is positive in Q1 and Q4
- So $ x = 53.1^\circ $ and $ 360^\circ - 53.1^\circ = 306.9^\circ $
- ✔ Answer: $ x = 53.1^\circ, 306.9^\circ $

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#### 6) $ \tan x = 0.2 $
- $ \tan^{-1}(0.2) \approx 11.3^\circ $
- Tangent is positive in Q1 and Q3
- So $ x = 11.3^\circ $ and $ 180^\circ + 11.3^\circ = 191.3^\circ $
- ✔ Answer: $ x = 11.3^\circ, 191.3^\circ $

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#### 7) $ 7\cos x = 3 $
- $ \cos x = \frac{3}{7} \approx 0.4286 $
- $ \cos^{-1}(0.4286) \approx 64.6^\circ $
- Positive in Q1 and Q4
- So $ x = 64.6^\circ $ and $ 360^\circ - 64.6^\circ = 295.4^\circ $
- ✔ Answer: $ x = 64.6^\circ, 295.4^\circ $

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#### 8) $ 2\tan x = \frac{1}{4} $
- $ \tan x = \frac{1}{8} = 0.125 $
- $ \tan^{-1}(0.125) \approx 7.1^\circ $
- Positive in Q1 and Q3
- So $ x = 7.1^\circ $ and $ 180^\circ + 7.1^\circ = 187.1^\circ $
- ✔ Answer: $ x = 7.1^\circ, 187.1^\circ $

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#### 9) $ 8 + 3\sin x = 10 $
- $ 3\sin x = 2 \Rightarrow \sin x = \frac{2}{3} \approx 0.6667 $
- $ \sin^{-1}(0.6667) \approx 41.8^\circ $
- Sine positive in Q1 and Q2
- So $ x = 41.8^\circ $ and $ 180^\circ - 41.8^\circ = 138.2^\circ $
- ✔ Answer: $ x = 41.8^\circ, 138.2^\circ $

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#### 10) $ \tan x = -1 $
- $ \tan^{-1}(-1) = -45^\circ $ → add 180° to get reference in range
- Tangent negative in Q2 and Q4
- Reference angle: $ 45^\circ $
- So $ x = 180^\circ - 45^\circ = 135^\circ $, $ x = 360^\circ - 45^\circ = 315^\circ $
- ✔ Answer: $ x = 135.0^\circ, 315.0^\circ $

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#### 11) $ \sin x = -0.9 $
- $ \sin^{-1}(0.9) \approx 64.2^\circ $
- Sine negative in Q3 and Q4
- So $ x = 180^\circ + 64.2^\circ = 244.2^\circ $, $ x = 360^\circ - 64.2^\circ = 295.8^\circ $
- ✔ Answer: $ x = 244.2^\circ, 295.8^\circ $

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#### 12) $ \tan x = -0.1 $
- $ \tan^{-1}(0.1) \approx 5.7^\circ $
- Tangent negative in Q2 and Q4
- So $ x = 180^\circ - 5.7^\circ = 174.3^\circ $, $ x = 360^\circ - 5.7^\circ = 354.3^\circ $
- ✔ Answer: $ x = 174.3^\circ, 354.3^\circ $

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#### 13) $ 7 + 8\sin x = 4 $
- $ 8\sin x = -3 \Rightarrow \sin x = -0.375 $
- $ \sin^{-1}(0.375) \approx 22.0^\circ $
- Sine negative in Q3 and Q4
- So $ x = 180^\circ + 22.0^\circ = 202.0^\circ $, $ x = 360^\circ - 22.0^\circ = 338.0^\circ $
- ✔ Answer: $ x = 202.0^\circ, 338.0^\circ $

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#### 14) $ \sin 2x = 0.2886 $
- Let $ y = 2x $. Then $ \sin y = 0.2886 $
- $ y = \sin^{-1}(0.2886) \approx 16.7^\circ $
- Also $ y = 180^\circ - 16.7^\circ = 163.3^\circ $
- But since $ y = 2x $, and $ x \in [0^\circ, 360^\circ] $, so $ y \in [0^\circ, 720^\circ] $
- So we find all solutions for $ y $ in $ [0^\circ, 720^\circ] $:
- $ y = 16.7^\circ, 163.3^\circ $
- Add $ 360^\circ $: $ 376.7^\circ, 523.3^\circ $
- Add another $ 360^\circ $: $ 736.7^\circ > 720^\circ $ → stop
- Now solve $ x = y/2 $:
- $ x = 16.7^\circ / 2 = 8.35^\circ \approx 8.4^\circ $
- $ x = 163.3^\circ / 2 = 81.65^\circ \approx 81.7^\circ $
- $ x = 376.7^\circ / 2 = 188.35^\circ \approx 188.4^\circ $
- $ x = 523.3^\circ / 2 = 261.65^\circ \approx 261.7^\circ $
- ✔ Answer: $ x = 8.4^\circ, 81.7^\circ, 188.4^\circ, 261.7^\circ $

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#### 15) $ \cos 3x = -0.3321 $
- Let $ y = 3x $, so $ \cos y = -0.3321 $
- $ \cos^{-1}(0.3321) \approx 70.7^\circ $
- Cosine negative in Q2 and Q3
- So $ y = 180^\circ - 70.7^\circ = 109.3^\circ $, $ y = 180^\circ + 70.7^\circ = 250.7^\circ $
- Now consider full period: $ y \in [0^\circ, 1080^\circ] $ (since $ x \leq 360^\circ \Rightarrow 3x \leq 1080^\circ $)
- Add $ 360^\circ $:
- $ 109.3^\circ + 360^\circ = 469.3^\circ $
- $ 250.7^\circ + 360^\circ = 610.7^\circ $
- Add again:
- $ 109.3^\circ + 720^\circ = 829.3^\circ $
- $ 250.7^\circ + 720^\circ = 970.7^\circ $
- Add again:
- $ 109.3 + 1080 = 1189.3^\circ > 1080^\circ $ → stop
- So valid $ y $ values: $ 109.3^\circ, 250.7^\circ, 469.3^\circ, 610.7^\circ, 829.3^\circ, 970.7^\circ $
- Now $ x = y / 3 $:
- $ x = 109.3 / 3 \approx 36.4^\circ $
- $ x = 250.7 / 3 \approx 83.6^\circ $
- $ x = 469.3 / 3 \approx 156.4^\circ $
- $ x = 610.7 / 3 \approx 203.6^\circ $
- $ x = 829.3 / 3 \approx 276.4^\circ $
- $ x = 970.7 / 3 \approx 323.6^\circ $
- ✔ Answer: $ x = 36.4^\circ, 83.6^\circ, 156.4^\circ, 203.6^\circ, 276.4^\circ, 323.6^\circ $

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#### 16) $ \sin(x + 20^\circ) = 0.6551 $
- Let $ y = x + 20^\circ $, so $ \sin y = 0.6551 $
- $ y = \sin^{-1}(0.6551) \approx 41.0^\circ $
- Also $ y = 180^\circ - 41.0^\circ = 139.0^\circ $
- Now $ y \in [20^\circ, 380^\circ] $ because $ x \in [0^\circ, 360^\circ] $
- So include:
- $ y = 41.0^\circ $ → $ x = 21.0^\circ $
- $ y = 139.0^\circ $ → $ x = 119.0^\circ $
- Add $ 360^\circ $: $ y = 41.0 + 360 = 401.0^\circ > 380^\circ $? No → skip
- $ y = 139.0 + 360 = 499^\circ $ → too big
- So only two solutions
- ✔ Answer: $ x = 21.0^\circ, 119.0^\circ $

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#### 17) $ \tan(x - 15^\circ) = -0.9128 $
- Let $ y = x - 15^\circ $, so $ \tan y = -0.9128 $
- $ \tan^{-1}(0.9128) \approx 42.3^\circ $
- Tangent negative in Q2 and Q4
- So $ y = 180^\circ - 42.3^\circ = 137.7^\circ $, $ y = 360^\circ - 42.3^\circ = 317.7^\circ $
- Also add $ 360^\circ $: $ 137.7 + 360 = 497.7^\circ $, $ 317.7 + 360 = 677.7^\circ $
- But $ y = x - 15^\circ $, $ x \in [0^\circ, 360^\circ] $ → $ y \in [-15^\circ, 345^\circ] $
- So valid $ y $: $ 137.7^\circ $, $ 317.7^\circ $
- $ x = y + 15^\circ $
- $ x = 137.7 + 15 = 152.7^\circ $
- $ x = 317.7 + 15 = 332.7^\circ $
- ✔ Answer: $ x = 152.7^\circ, 332.7^\circ $

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#### 18) $ \cos(2x + 33^\circ) = 0.306 $
- Let $ y = 2x + 33^\circ $, so $ \cos y = 0.306 $
- $ \cos^{-1}(0.306) \approx 72.3^\circ $
- Cosine positive in Q1 and Q4
- So $ y = 72.3^\circ $, $ y = 360^\circ - 72.3^\circ = 287.7^\circ $
- Now $ y \in [33^\circ, 753^\circ] $ (since $ x \in [0^\circ, 360^\circ] $ → $ 2x \in [0^\circ, 720^\circ] $ → $ y \in [33^\circ, 753^\circ] $)
- Add $ 360^\circ $: $ 72.3 + 360 = 432.3^\circ $, $ 287.7 + 360 = 647.7^\circ $
- Add again: $ 72.3 + 720 = 792.3^\circ > 753^\circ $ → stop
- So valid $ y $: $ 72.3^\circ, 287.7^\circ, 432.3^\circ, 647.7^\circ $
- Solve for $ x $: $ x = \frac{y - 33^\circ}{2} $
- $ x = (72.3 - 33)/2 = 19.65 \approx 19.7^\circ $
- $ x = (287.7 - 33)/2 = 127.35 \approx 127.4^\circ $
- $ x = (432.3 - 33)/2 = 199.65 \approx 199.7^\circ $
- $ x = (647.7 - 33)/2 = 307.35 \approx 307.4^\circ $
- ✔ Answer: $ x = 19.7^\circ, 127.4^\circ, 199.7^\circ, 307.4^\circ $

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#### 19) $ \tan^2 x = \frac{1}{4} $
- $ \tan x = \pm \frac{1}{2} $
- Case 1: $ \tan x = 0.5 $ → $ x = \tan^{-1}(0.5) \approx 26.6^\circ $
- In Q1 and Q3: $ x = 26.6^\circ, 206.6^\circ $
- Case 2: $ \tan x = -0.5 $ → $ \tan^{-1}(0.5) = 26.6^\circ $
- Negative in Q2 and Q4: $ x = 180^\circ - 26.6^\circ = 153.4^\circ $, $ x = 360^\circ - 26.6^\circ = 333.4^\circ $
- ✔ Answer: $ x = 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ $

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#### 20) $ \cos^2 x = \frac{1}{2} $
- $ \cos x = \pm \frac{\sqrt{2}}{2} \approx \pm 0.7071 $
- Case 1: $ \cos x = 0.7071 $ → $ x = \cos^{-1}(0.7071) \approx 45^\circ $
- Positive in Q1 and Q4: $ x = 45^\circ, 315^\circ $
- Case 2: $ \cos x = -0.7071 $ → $ x = \cos^{-1}(0.7071) = 45^\circ $
- Negative in Q2 and Q3: $ x = 180^\circ - 45^\circ = 135^\circ $, $ x = 180^\circ + 45^\circ = 225^\circ $
- ✔ Answer: $ x = 45.0^\circ, 135.0^\circ, 225.0^\circ, 315.0^\circ $

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✔ Final Answers (Rounded to Nearest Tenth):

| Problem | Answer(s) |
|--------|-----------|
| 1 | $ 90.0^\circ $ |
| 2 | $ 45.0^\circ, 225.0^\circ $ |
| 3 | $ 0.0^\circ, 360.0^\circ $ |
| 4 | $ 30.0^\circ, 150.0^\circ $ |
| 5 | $ 53.1^\circ, 306.9^\circ $ |
| 6 | $ 11.3^\circ, 191.3^\circ $ |
| 7 | $ 64.6^\circ, 295.4^\circ $ |
| 8 | $ 7.1^\circ, 187.1^\circ $ |
| 9 | $ 41.8^\circ, 138.2^\circ $ |
| 10 | $ 135.0^\circ, 315.0^\circ $ |
| 11 | $ 244.2^\circ, 295.8^\circ $ |
| 12 | $ 174.3^\circ, 354.3^\circ $ |
| 13 | $ 202.0^\circ, 338.0^\circ $ |
| 14 | $ 8.4^\circ, 81.7^\circ, 188.4^\circ, 261.7^\circ $ |
| 15 | $ 36.4^\circ, 83.6^\circ, 156.4^\circ, 203.6^\circ, 276.4^\circ, 323.6^\circ $ |
| 16 | $ 21.0^\circ, 119.0^\circ $ |
| 17 | $ 152.7^\circ, 332.7^\circ $ |
| 18 | $ 19.7^\circ, 127.4^\circ, 199.7^\circ, 307.4^\circ $ |
| 19 | $ 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ $ |
| 20 | $ 45.0^\circ, 135.0^\circ, 225.0^\circ, 315.0^\circ $ |

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