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# AS-Level Maths | Trigonometry
- URL: https://www.a-level-maths-tutor.com/as-level-maths-trigonometry/
- Published: 2026-04-23T12:54:17.000Z
- Updated: 2026-06-25T10:33:23.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

**Worked Example**

$\\sin x° = 0.4$, $0 \\leq x \\leq 360°$

$x = \\sin^{-1}(0.4)$

$x = 23.6°, 156.4°$

## Question 1

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\sin x° = 0.27$

**b)** $\\sin x° = 0.813$

**c)** $\\sin x° = 0.6$

**d)** $\\sin x° = 0.95$

Show Answers 

**a)** $x = 15.7°, 164.3°$ **b)** $x = 54.4°, 125.6°$ **c)** $x = 36.9°, 143.1°$ **d)** $x = 71.8°, 108.2°$

**Worked Example**

$\\cos x° = 0.6$, $0 \\leq x \\leq 360°$

$x = \\cos^{-1}(0.6)$

$x = 53.1°, 306.9°$

## Question 2

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\cos x° = 0.185$

**b)** $\\cos x° = 0.4$

**c)** $\\cos x° = 0.7$

**d)** $\\cos x° = 0.92$

Show Answers 

**a)** $x = 79.3°, 280.7°$ **b)** $x = 66.4°, 293.6°$ **c)** $x = 45.6°, 314.4°$ **d)** $x = 23.1°, 336.9°$

**Worked Example**

$\\tan x° = 0.8$, $0 \\leq x \\leq 360°$

$x = \\tan^{-1}(0.8)$

$x = 38.7°, 218.7°$

## Question 3

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\tan x° = 0.1$

**b)** $\\tan x° = 1.6$

**c)** $\\tan x° = 0.7$

**d)** $\\tan x° = 3.4$

Show Answers 

**a)** $x = 5.7°, 185.7°$ **b)** $x = 58.0°, 238.0°$ **c)** $x = 35.0°, 215.0°$ **d)** $x = 73.6°, 253.6°$

After finding two consecutive solutions, keep adding or subtracting $360°$ to each solution to find more solutions.

e.g. $\\ldots, -330°, -210°, 30°, 150°, 390°, 510°, \\ldots$

## Question 4

Solve each equation for $x$ in the interval $0 \\leq x \\leq 720°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\sin x° = 0.3$

**b)** $\\sin x° = 0.85$

**c)** $\\sin x° = -0.45$

**d)** $\\sin x° = -0.7$

Show Answers 

**a)** $x = 17.5°, 162.5°, 377.5°, 522.5°$

**b)** $x = 58.2°, 121.8°, 418.2°, 481.8°$

**c)** $x = 206.7°, 333.3°, 566.7°, 693.3°$

**d)** $x = 224.4°, 315.6°, 584.4°, 675.6°$

## Question 5

Solve each equation for $x$ in the interval $0 \\leq x \\leq 720°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\cos x° = -0.39$

**b)** $\\cos x° = -0.636$

**c)** $\\cos x° = 0.55$

**d)** $\\cos x° = -0.8$

Show Answers 

**a)** $x = 113.0°, 247.1°, 473.0°, 607.1°$

**b)** $x = 129.5°, 230.5°, 489.5°, 590.5°$

**c)** $x = 56.6°, 303.4°, 416.6°, 663.4°$

**d)** $x = 143.1°, 216.9°, 503.1°, 576.9°$

## Question 6

Solve each equation for $x$ in the interval $0 \\leq x \\leq 720°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\tan x° = -0.7$

**b)** $\\tan x° = -3.4$

**c)** $\\tan x° = 2.1$

**d)** $\\tan x° = -1.9$

Show Answers 

**a)** $x = 145.0°, 325.0°, 505.0°, 685.0°$

**b)** $x = 106.4°, 286.4°, 466.4°, 646.4°$

**c)** $x = 64.5°, 244.5°, 424.5°, 604.5°$

**d)** $x = 117.8°, 297.8°, 477.8°, 657.8°$

**Adjusting the Range**

If the original range is $0° \\leq x \\leq 360°$, this changes depending on what's inside the brackets:

$\\sin(x-30)°$: range becomes $-30° \\leq x-30 \\leq 330°$

$\\sin(x+45)°$: range becomes $45° \\leq x+45 \\leq 405°$

$\\sin 2x°$: range becomes $0° \\leq 2x \\leq 720°$

$\\sin 3x°$: range becomes $0° \\leq 3x \\leq 1080°$

**Worked Example**

$\\sin(x-30)° = 0.5$, $0 \\leq x \\leq 360°$

$x - 30 = \\sin^{-1}(0.5)$

$x - 30 = 30°, 150°$

$x = 60°, 180°$

## Question 7

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\sin(x-30)° = 0.5$

**b)** $\\sin(x-45)° = 0.6$

**c)** $\\sin(x-60)° = -0.4$

**d)** $\\sin(x-20)° = -0.75$

Show Answers 

**a)** $x = 60°, 180°$ **b)** $x = 81.9°, 188.1°$ **c)** $x = 36.4°, 263.6°$ **d)** $x = 248.6°, 331.4°$

## Question 8

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\sin(x+30)° = 0.7$

**b)** $\\sin(x+40)° = 0.3$

**c)** $\\sin(x+50)° = -0.6$

**d)** $\\sin(x+25)° = -0.45$

Show Answers 

**a)** $x = 14.4°, 105.6°$ **b)** $x = 122.5°, 337.5°$ **c)** $x = 166.9°, 273.1°$ **d)** $x = 181.7°, 308.3°$

## Question 9

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\sin 2x° = 0.5$

**b)** $\\sin 2x° = -0.7$

**c)** $\\sin 3x° = 0.6$

**d)** $\\sin 3x° = -0.3$

Show Answers 

**a)** $x = 15°, 75°, 195°, 255°$

**b)** $x = 112.2°, 157.8°, 292.2°, 337.8°$

**c)** $x = 12.3°, 47.7°, 132.3°, 167.7°, 252.3°, 287.7°$

**d)** $x = 65.8°, 114.2°, 185.8°, 234.2°, 305.8°, 354.2°$

## Question 10

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\sin \\tfrac{1}{2}x° = 0.5$

**b)** $\\sin \\tfrac{1}{2}x° = 0.8$

**c)** $\\sin \\tfrac{1}{3}x° = 0.6$

**d)** $\\sin \\tfrac{1}{3}x° = 0.85$

Show Answers 

**a)** $x = 60°, 300°$ **b)** $x = 106.3°, 253.7°$ **c)** $x = 110.6°$ **d)** $x = 174.6°$

$\\cos(x+30)° = 0.5$, $0 \\leq x \\leq 360°$

$x + 30 = 60°, 300°$

$x = 30°, 270°$

## Question 11

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\cos(x-30)° = 0.5$

**b)** $\\cos(x-45)° = -0.6$

**c)** $\\cos(x-60)° = 0.4$

**d)** $\\cos(x-20)° = -0.75$

Show Answers 

**a)** $x = 90°, 330°$ **b)** $x = 171.9°, 278.1°$ **c)** $x = 126.4°, 353.6°$ **d)** $x = 158.6°, 241.4°$

## Question 12

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\cos(x+30)° = 0.5$

**b)** $\\cos(x+40)° = -0.3$

**c)** $\\cos(x+50)° = 0.6$

**d)** $\\cos(x+25)° = -0.45$

Show Answers 

**a)** $x = 30°, 270°$ **b)** $x = 67.5°, 212.5°$ **c)** $x = 3.1°, 256.9°$ **d)** $x = 91.7°, 218.3°$

## Question 13

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\cos 2x° = 0.64$

**b)** $\\cos 2x° = -0.415$

**c)** $\\cos 3x° = 0.5$

**d)** $\\cos 3x° = -0.2$

Show Answers 

**a)** $x = 25.1°, 154.9°, 205.1°, 334.9°$

**b)** $x = 57.3°, 122.7°, 237.3°, 302.7°$

**c)** $x = 20°, 100°, 140°, 220°, 260°, 340°$

**d)** $x = 33.9°, 86.2°, 153.9°, 206.2°, 273.9°, 326.2°$

## Question 14

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\cos \\tfrac{1}{2}x° = 0.5$

**b)** $\\cos \\tfrac{1}{2}x° = -0.6$

**c)** $\\cos \\tfrac{1}{3}x° = 0.7$

**d)** $\\cos \\tfrac{1}{3}x° = -0.4$

Show Answers 

**a)** $x = 120°$ **b)** $x = 253.7°$ **c)** $x = 136.7°$ **d)** $x = 340.7°$

$\\tan(x-30)° = 1$, $0 \\leq x \\leq 360°$

$x - 30 = 45°, 225°$

$x = 75°, 255°$

## Question 15

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\tan(x-30)° = 1$

**b)** $\\tan(x-45)° = -1.26$

**c)** $\\tan(x-60)° = 0.5$

**d)** $\\tan(x-20)° = -2.74$

Show Answers 

**a)** $x = 75°, 255°$ **b)** $x = 173.4°, 353.4°$ **c)** $x = 86.6°, 266.6°$ **d)** $x = 130.1°, 310.1°$

## Question 16

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\tan(x+30)° = 1$

**b)** $\\tan(x+40)° = -1.26$

**c)** $\\tan(x+50)° = 0.591$

**d)** $\\tan(x+25)° = -2.74$

Show Answers 

**a)** $x = 15°, 195°$ **b)** $x = 88.4°, 268.4°$ **c)** $x = 160.6°, 340.6°$ **d)** $x = 85.1°, 265.1°$

## Question 17

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\tan 2x° = -2.74$

**b)** $\\tan 2x° = 0.591$

**c)** $\\tan 3x° = 1$

**d)** $\\tan 3x° = -1.26$

Show Answers 

**a)** $x = 55.0°, 145.0°, 235.0°, 325.0°$

**b)** $x = 15.3°, 105.3°, 195.3°, 285.3°$

**c)** $x = 15°, 75°, 135°, 195°, 255°, 315°$

**d)** $x = 42.8°, 102.8°, 162.8°, 222.8°, 282.8°, 342.8°$

## Question 18

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\tan \\tfrac{1}{2}x° = 1$

**b)** $\\tan \\tfrac{1}{2}x° = -1.26$

**c)** $\\tan \\tfrac{1}{3}x° = 0.591$

**d)** $\\tan \\tfrac{1}{3}x° = -2.74$

Show Answers 

**a)** $x = 90°$ **b)** $x = 256.9°$ **c)** $x = 91.7°$ **d)** $x = 330.2°$

$2\\sin^2\\theta° - 3\\sin\\theta° + 1 = 0$, $0 \\leq \\theta \\leq 360°$

$(2\\sin\\theta°-1)(\\sin\\theta°-1) = 0$

$\\sin\\theta° = \\tfrac{1}{2}, 1$

$\\theta = 30°, 90°, 150°$

## Question 19

Solve each equation for $\\theta$ in the interval $0 \\leq \\theta \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $\\sin^2\\theta° = 0.75$

**b)** $1 - \\tan^2\\theta° = 0$

**c)** $2\\cos^2\\theta° + \\cos\\theta° = 0$

**d)** $\\sin\\theta°(4\\cos\\theta° - 1) = 0$

Show Answers 

**a)** $\\theta = 60°, 120°, 240°, 300°$ **b)** $\\theta = 45°, 135°, 225°, 315°$

**c)** $\\theta = 90°, 120°, 240°, 270°$ **d)** $\\theta = 0°, 75.5°, 180°, 284.5°, 360°$

$\\sin^2\\theta + \\cos^2\\theta = 1$

$\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}$

$2\\sin^2 x - 5\\cos x + 1 = 0$, $0 \\leq x \\leq 360°$

$2(1-\\cos^2 x) - 5\\cos x + 1 = 0$

$2\\cos^2 x + 5\\cos x - 3 = 0$

$(2\\cos x - 1)(\\cos x + 3) = 0$

$\\cos x = \\tfrac{1}{2}$

$x = 60°, 300°$

## Question 20

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $2\\sin^2 x - 5\\cos x + 1 = 0$

**b)** $5\\sin^2 x = 19\\cos x + 1$

**c)** $3\\sin^2 x = 8 + 16\\cos x$

**d)** $\\tan^2 x - \\tan x = 6$

**e)** $6\\cos^2 x - \\cos x - 2 = 0$

Show Answers 

**a)** $x = 60°, 300°$ **b)** $x \\approx 78.5°, 281.5°$ **c)** $x \\approx 109.5°, 250.5°$

**d)** $x \\approx 71.6°, 116.6°, 251.6°, 296.6°$ **e)** $x \\approx 48.2°, 120°, 240°, 311.8°$

$3\\cos^2 x - \\sin^2 x = 2$, $0 \\leq x \\leq 360°$

$3\\cos^2 x - (1-\\cos^2 x) = 2$

$4\\cos^2 x = 3$

$\\cos x = \\pm\\dfrac{\\sqrt{3}}{2}$

$x = 30°, 150°, 210°, 330°$

## Question 21

Solve each equation for $x$ in the interval $0 \\leq x \\leq 360°$. Give your answers to 1 decimal place where appropriate.

**a)** $3\\cos x = 8\\tan x$

**b)** $2\\tan x \\sin x = 3$

**c)** $2\\cos x = 3\\tan x$

**d)** $3\\sin x - 2\\tan x = 0$

**e)** $\\cos x = 3\\tan x$

Show Answers 

**a)** $x \\approx 19.5°, 160.5°$ **b)** $x = 60°, 300°$ **c)** $x = 30°, 150°$

**d)** $x = 0°, 48.2°, 180°, 311.8°, 360°$ **e)** $x \\approx 17.6°, 162.4°$

## Question 22

Solve each of the following trigonometric equations in the range given.

**a)** $4\\tan^2\\theta\\cos\\theta = 15$ $0 \\leq \\theta < 360°$

**b)** $\\dfrac{3 + \\sin^2\\theta}{\\cos\\theta - 2} = 3\\cos\\theta$ $0 \\leq \\theta < 360°$

**c)** $2\\sin x = \\tan x$ $0 \\leq x < 360°$

**d)** $\\tfrac{1}{2}\\tan x - \\sin x = 0$ $0 \\leq x < 360°$

**e)** $\\dfrac{\\sin x - \\cos x}{\\cos x} = 2$ $0 \\leq x < 360°$

**f)** $4\\tan y\\sin y\\cos y + 4\\tan y\\cos y + 1 = 0$ $0 \\leq y < 360°$

Show Answers 

**a)** $\\theta \\approx 75.5°, 284.5°$ **b)** $\\theta = 120°, 240°$ **c)** $x = 0°, 60°, 180°, 300°$

**d)** $x = 0°, 60°, 180°, 300°$ **e)** $x \\approx 71.6°, 251.6°$ **f)** $y = 210°, 330°$

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