Solving Trigonometric Equations worksheet - Free Printable

Here are the solutions for the trigonometric equations. I have calculated the values using a calculator and applied the rules for finding angles in different quadrants (0° to 360°). All answers are rounded to 1 decimal place as requested.

1. $\sin \theta = \frac{1}{3}$
* Calculator value: $\sin^{-1}(1/3) \approx 19.47^\circ$
* Since sine is positive, angles are in Quadrant I and II.
* Q1: $19.5^\circ$
* Q2: $180^\circ - 19.47^\circ = 160.53^\circ \rightarrow 160.5^\circ$

2. $\cos \theta = 0.507$
* Calculator value: $\cos^{-1}(0.507) \approx 59.54^\circ$
* Since cosine is positive, angles are in Quadrant I and IV.
* Q1: $59.5^\circ$
* Q4: $360^\circ - 59.54^\circ = 300.46^\circ \rightarrow 300.5^\circ$

3. $\tan \theta = 1.15$
* Calculator value: $\tan^{-1}(1.15) \approx 48.99^\circ$
* Since tangent is positive, angles are in Quadrant I and III.
* Q1: $49.0^\circ$
* Q3: $180^\circ + 48.99^\circ = 228.99^\circ \rightarrow 229.0^\circ$

4. $\cos \theta = -0.15$
* Calculator value: $\cos^{-1}(-0.15) \approx 98.63^\circ$ (This gives the Q2 angle directly)
* Since cosine is negative, angles are in Quadrant II and III.
* Q2: $98.6^\circ$
* Q3: $360^\circ - 98.63^\circ = 261.37^\circ \rightarrow 261.4^\circ$

5. $\sin \theta = -0.25$
* Calculator value: $\sin^{-1}(-0.25) \approx -14.48^\circ$ (Reference angle is $14.48^\circ$)
* Since sine is negative, angles are in Quadrant III and IV.
* Q3: $180^\circ + 14.48^\circ = 194.48^\circ \rightarrow 194.5^\circ$
* Q4: $360^\circ - 14.48^\circ = 345.52^\circ \rightarrow 345.5^\circ$

6. $\tan \theta = -0.75$
* Calculator value: $\tan^{-1}(-0.75) \approx -36.87^\circ$ (Reference angle is $36.87^\circ$)
* Since tangent is negative, angles are in Quadrant II and IV.
* Q2: $180^\circ - 36.87^\circ = 143.13^\circ \rightarrow 143.1^\circ$
* Q4: $360^\circ - 36.87^\circ = 323.13^\circ \rightarrow 323.1^\circ$

7. $\sin \theta = -0.1178$
* Calculator value: $\sin^{-1}(-0.1178) \approx -6.77^\circ$ (Reference angle is $6.77^\circ$)
* Since sine is negative, angles are in Quadrant III and IV.
* Q3: $180^\circ + 6.77^\circ = 186.77^\circ \rightarrow 186.8^\circ$
* Q4: $360^\circ - 6.77^\circ = 353.23^\circ \rightarrow 353.2^\circ$

8. $\sin \theta = -0.55$
* Calculator value: $\sin^{-1}(-0.55) \approx -33.37^\circ$ (Reference angle is $33.37^\circ$)
* Since sine is negative, angles are in Quadrant III and IV.
* Q3: $180^\circ + 33.37^\circ = 213.37^\circ \rightarrow 213.4^\circ$
* Q4: $360^\circ - 33.37^\circ = 326.63^\circ \rightarrow 326.6^\circ$

9. $\cos \theta = 0.23$
* Calculator value: $\cos^{-1}(0.23) \approx 76.70^\circ$
* Since cosine is positive, angles are in Quadrant I and IV.
* Q1: $76.7^\circ$
* Q4: $360^\circ - 76.70^\circ = 283.30^\circ \rightarrow 283.3^\circ$

10. $\tan \theta = -1.07$
* Calculator value: $\tan^{-1}(-1.07) \approx -46.95^\circ$ (Reference angle is $46.95^\circ$)
* Since tangent is negative, angles are in Quadrant II and IV.
* Q2: $180^\circ - 46.95^\circ = 133.05^\circ \rightarrow 133.1^\circ$
* Q4: $360^\circ - 46.95^\circ = 313.05^\circ \rightarrow 313.1^\circ$

Final Answer:
1. θ = 19.5°, θ = 160.5°
2. θ = 59.5°, θ = 300.5°
3. θ = 49.0°, θ = 229.0°
4. θ = 98.6°, θ = 261.4°
5. θ = 194.5°, θ = 345.5°
6. θ = 143.1°, θ = 323.1°
7. θ = 186.8°, θ = 353.2°
8. θ = 213.4°, θ = 326.6°
9. θ = 76.7°, θ = 283.3°
10. θ = 133.1°, θ = 313.1°