Trigonometric equations worksheet for GCSE level, solving for x in degrees.

To solve the trigonometric equations for $0^\circ \le x \le 360^\circ$, we need to find all angles within this range that satisfy each equation. We'll use inverse trigonometric functions and consider the periodic nature of sine, cosine, and tangent.

Let's go through each problem step by step:

1) $\sin x = 1$
- The sine function equals 1 at $x = 90^\circ$.
- Final Answer: $x = 90.0^\circ$

2) $\tan x = 1$
- The tangent function equals 1 at $x = 45^\circ$ and $x = 225^\circ$ (since $\tan(180^\circ + \theta) = \tan \theta$).
- Final Answer: $x = 45.0^\circ, 225.0^\circ$

3) $\cos x = 1$
- The cosine function equals 1 at $x = 0^\circ$ and $x = 360^\circ$.
- Final Answer: $x = 0.0^\circ, 360.0^\circ$

4) $\sin x = 0.5$
- The sine function equals 0.5 at $x = 30^\circ$ and $x = 150^\circ$ (since $\sin(180^\circ - \theta) = \sin \theta$).
- Final Answer: $x = 30.0^\circ, 150.0^\circ$

5) $\cos x = 0.6$
- Using a calculator, $\cos^{-1}(0.6) \approx 53.1^\circ$. Since cosine is positive in the first and fourth quadrants, the solutions are $x = 53.1^\circ$ and $x = 360^\circ - 53.1^\circ = 306.9^\circ$.
- Final Answer: $x = 53.1^\circ, 306.9^\circ$

6) $\tan x = 0.2$
- Using a calculator, $\tan^{-1}(0.2) \approx 11.3^\circ$. Since tangent is positive in the first and third quadrants, the solutions are $x = 11.3^\circ$ and $x = 180^\circ + 11.3^\circ = 191.3^\circ$.
- Final Answer: $x = 11.3^\circ, 191.3^\circ$

7) $7 \cos x = 3$
- Solving for $\cos x$, we get $\cos x = \frac{3}{7} \approx 0.4286$. Using a calculator, $\cos^{-1}(0.4286) \approx 64.6^\circ$. Since cosine is positive in the first and fourth quadrants, the solutions are $x = 64.6^\circ$ and $x = 360^\circ - 64.6^\circ = 295.4^\circ$.
- Final Answer: $x = 64.6^\circ, 295.4^\circ$

8) $2 \tan x = \frac{1}{4}$
- Solving for $\tan x$, we get $\tan x = \frac{1}{8} = 0.125$. Using a calculator, $\tan^{-1}(0.125) \approx 7.1^\circ$. Since tangent is positive in the first and third quadrants, the solutions are $x = 7.1^\circ$ and $x = 180^\circ + 7.1^\circ = 187.1^\circ$.
- Final Answer: $x = 7.1^\circ, 187.1^\circ$

9) $8 + 3 \sin x = 10$
- Solving for $\sin x$, we get $\sin x = \frac{2}{3} \approx 0.6667$. Using a calculator, $\sin^{-1}(0.6667) \approx 41.8^\circ$. Since sine is positive in the first and second quadrants, the solutions are $x = 41.8^\circ$ and $x = 180^\circ - 41.8^\circ = 138.2^\circ$.
- Final Answer: $x = 41.8^\circ, 138.2^\circ$

10) $\tan x = -1$
- The tangent function equals -1 at $x = 135^\circ$ and $x = 315^\circ$ (since $\tan(180^\circ - \theta) = -\tan \theta$ and $\tan(360^\circ - \theta) = -\tan \theta$).
- Final Answer: $x = 135.0^\circ, 315.0^\circ$

11) $\sin x = -0.9$
- Using a calculator, $\sin^{-1}(-0.9) \approx -64.2^\circ$. Since sine is negative in the third and fourth quadrants, the solutions are $x = 180^\circ + 64.2^\circ = 244.2^\circ$ and $x = 360^\circ - 64.2^\circ = 295.8^\circ$.
- Final Answer: $x = 244.2^\circ, 295.8^\circ$

12) $\tan x = -0.1$
- Using a calculator, $\tan^{-1}(-0.1) \approx -5.7^\circ$. Since tangent is negative in the second and fourth quadrants, the solutions are $x = 180^\circ - 5.7^\circ = 174.3^\circ$ and $x = 360^\circ - 5.7^\circ = 354.3^\circ$.
- Final Answer: $x = 174.3^\circ, 354.3^\circ$

13) $7 + 8 \sin x = 4$
- Solving for $\sin x$, we get $\sin x = -\frac{3}{8} = -0.375$. Using a calculator, $\sin^{-1}(-0.375) \approx -22.0^\circ$. Since sine is negative in the third and fourth quadrants, the solutions are $x = 180^\circ + 22.0^\circ = 202.0^\circ$ and $x = 360^\circ - 22.0^\circ = 338.0^\circ$.
- Final Answer: $x = 202.0^\circ, 338.0^\circ$

14) $\sin 2x = 0.2886$
- Let $y = 2x$. Then $\sin y = 0.2886$. Using a calculator, $\sin^{-1}(0.2886) \approx 16.8^\circ$. So $y = 16.8^\circ$ or $y = 180^\circ - 16.8^\circ = 163.2^\circ$. Therefore, $2x = 16.8^\circ$ or $2x = 163.2^\circ$, which gives $x = 8.4^\circ$ or $x = 81.6^\circ$. Also, considering the periodicity, $2x = 360^\circ + 16.8^\circ = 376.8^\circ$ or $2x = 360^\circ + 163.2^\circ = 523.2^\circ$, which gives $x = 188.4^\circ$ or $x = 261.6^\circ$.
- Final Answer: $x = 8.4^\circ, 81.6^\circ, 188.4^\circ, 261.6^\circ$

15) $\cos 3x = -0.3321$
- Let $y = 3x$. Then $\cos y = -0.3321$. Using a calculator, $\cos^{-1}(-0.3321) \approx 109.4^\circ$. So $y = 109.4^\circ$ or $y = 360^\circ - 109.4^\circ = 250.6^\circ$. Therefore, $3x = 109.4^\circ$ or $3x = 250.6^\circ$, which gives $x = 36.5^\circ$ or $x = 83.5^\circ$. Also, considering the periodicity, $3x = 360^\circ + 109.4^\circ = 469.4^\circ$ or $3x = 360^\circ + 250.6^\circ = 610.6^\circ$, which gives $x = 156.5^\circ$ or $x = 203.5^\circ$. Finally, $3x = 720^\circ + 109.4^\circ = 829.4^\circ$ or $3x = 720^\circ + 250.6^\circ = 970.6^\circ$, which gives $x = 276.5^\circ$ or $x = 323.5^\circ$.
- Final Answer: $x = 36.5^\circ, 83.5^\circ, 156.5^\circ, 203.5^\circ, 276.5^\circ, 323.5^\circ$

16) $\sin(x + 20^\circ) = 0.6551$
- Let $y = x + 20^\circ$. Then $\sin y = 0.6551$. Using a calculator, $\sin^{-1}(0.6551) \approx 41.0^\circ$. So $y = 41.0^\circ$ or $y = 180^\circ - 41.0^\circ = 139.0^\circ$. Therefore, $x + 20^\circ = 41.0^\circ$ or $x + 20^\circ = 139.0^\circ$, which gives $x = 21.0^\circ$ or $x = 119.0^\circ$. Also, considering the periodicity, $y = 360^\circ + 41.0^\circ = 401.0^\circ$ or $y = 360^\circ + 139.0^\circ = 499.0^\circ$, which gives $x = 381.0^\circ - 20^\circ = 361.0^\circ$ (not in range) or $x = 479.0^\circ - 20^\circ = 459.0^\circ$ (not in range).
- Final Answer: $x = 21.0^\circ, 119.0^\circ$

17) $\tan(x - 15^\circ) = -0.9128$
- Let $y = x - 15^\circ$. Then $\tan y = -0.9128$. Using a calculator, $\tan^{-1}(-0.9128) \approx -42.4^\circ$. So $y = -42.4^\circ$ or $y = 180^\circ - 42.4^\circ = 137.6^\circ$. Therefore, $x - 15^\circ = -42.4^\circ$ or $x - 15^\circ = 137.6^\circ$, which gives $x = -27.4^\circ + 360^\circ = 332.6^\circ$ or $x = 152.6^\circ$. Also, considering the periodicity, $y = 180^\circ - 42.4^\circ = 137.6^\circ$ (already considered), so no additional solutions.
- Final Answer: $x = 152.6^\circ, 332.6^\circ$

18) $\cos(2x + 33^\circ) = 0.306$
- Let $y = 2x + 33^\circ$. Then $\cos y = 0.306$. Using a calculator, $\cos^{-1}(0.306) \approx 72.2^\circ$. So $y = 72.2^\circ$ or $y = 360^\circ - 72.2^\circ = 287.8^\circ$. Therefore, $2x + 33^\circ = 72.2^\circ$ or $2x + 33^\circ = 287.8^\circ$, which gives $2x = 39.2^\circ$ or $2x = 254.8^\circ$, so $x = 19.6^\circ$ or $x = 127.4^\circ$. Also, considering the periodicity, $y = 360^\circ + 72.2^\circ = 432.2^\circ$ or $y = 360^\circ + 287.8^\circ = 647.8^\circ$, which gives $2x = 399.2^\circ$ or $2x = 614.8^\circ$, so $x = 199.6^\circ$ or $x = 307.4^\circ$.
- Final Answer: $x = 19.6^\circ, 127.4^\circ, 199.6^\circ, 307.4^\circ$

19) $\tan^2 x = \frac{1}{4}$
- Taking the square root, $\tan x = \pm \frac{1}{2}$. For $\tan x = \frac{1}{2}$, using a calculator, $\tan^{-1}\left(\frac{1}{2}\right) \approx 26.6^\circ$. So $x = 26.6^\circ$ or $x = 180^\circ + 26.6^\circ = 206.6^\circ$. For $\tan x = -\frac{1}{2}$, using a calculator, $\tan^{-1}\left(-\frac{1}{2}\right) \approx -26.6^\circ$. So $x = 180^\circ - 26.6^\circ = 153.4^\circ$ or $x = 360^\circ - 26.6^\circ = 333.4^\circ$.
- Final Answer: $x = 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ$

20) $\cos^2 x = \frac{1}{2}$
- Taking the square root, $\cos x = \pm \frac{\sqrt{2}}{2}$. For $\cos x = \frac{\sqrt{2}}{2}$, $x = 45^\circ$ or $x = 360^\circ - 45^\circ = 315^\circ$. For $\cos x = -\frac{\sqrt{2}}{2}$, $x = 180^\circ - 45^\circ = 135^\circ$ or $x = 180^\circ + 45^\circ = 225^\circ$.
- Final Answer: $x = 45.0^\circ, 135.0^\circ, 225.0^\circ, 315.0^\circ$

Final Answer:
1) $x = 90.0^\circ$
2) $x = 45.0^\circ, 225.0^\circ$
3) $x = 0.0^\circ, 360.0^\circ$
4) $x = 30.0^\circ, 150.0^\circ$
5) $x = 53.1^\circ, 306.9^\circ$
6) $x = 11.3^\circ, 191.3^\circ$
7) $x = 64.6^\circ, 295.4^\circ$
8) $x = 7.1^\circ, 187.1^\circ$
9) $x = 41.8^\circ, 138.2^\circ$
10) $x = 135.0^\circ, 315.0^\circ$
11) $x = 244.2^\circ, 295.8^\circ$
12) $x = 174.3^\circ, 354.3^\circ$
13) $x = 202.0^\circ, 338.0^\circ$
14) $x = 8.4^\circ, 81.6^\circ, 188.4^\circ, 261.6^\circ$
15) $x = 36.5^\circ, 83.5^\circ, 156.5^\circ, 203.5^\circ, 276.5^\circ, 323.5^\circ$
16) $x = 21.0^\circ, 119.0^\circ$
17) $x = 152.6^\circ, 332.6^\circ$
18) $x = 19.6^\circ, 127.4^\circ, 199.6^\circ, 307.4^\circ$
19) $x = 26.6^\circ, 153.4^\circ, 206.6^\circ, 333.4^\circ$
20) $x = 45.0^\circ, 135.0^\circ, 225.0^\circ, 315.0^\circ$