AS-Level Maths | Trigonometry
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) $\sin x° = 0.95$
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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) $\tan x° = 3.4$
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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) $\sin x° = -0.7$
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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) $\tan x° = -1.9$
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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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