Sec 3 · Additional Mathematics

Trigonometric equations

Practise trigonometric equations with step-by-step help. Free account.

1. Principal values

Function	Principal value	In radians	Example
$\sin^{-1}$	$-90^\circ \leq \theta \leq 90^\circ$	$-\dfrac{\pi}{2} \leq \theta \leq \dfrac{\pi}{2}$	$\sin^{-1}\left(-\dfrac{1}{2}\right) = -30^\circ$
$\cos^{-1}$	$0^\circ \leq \theta \leq 180^\circ$	$0 \leq \theta \leq \pi$	$\cos^{-1}\left(-\dfrac{1}{2}\right) = 120^\circ$
$\tan^{-1}$	$-90^\circ < \theta < 90^\circ$	$-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$	$\tan^{-1}(-1) = -45^\circ$

📖 Worked example

Without using a calculator, find the principal value, in degrees, of $\cos^{-1}\left(-\dfrac{1}{2}\right)$ .

Step 1 of 3

Find the basic angle (use the positive value)

\text{Basic angle} = 60^\circ

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2. Solving sin x = k, cos x = k, tan x = k

ASTC + basic angle	Ch 9: angle → value	Ch 10: value → angles
given	$\cos 200^\circ$	$\cos x = -0.6$
1. basic angle $\alpha$	from the angle: $200^\circ - 180^\circ = 20^\circ$	from the positive value: $\cos^{-1} 0.6 = 53.1^\circ$
2. ASTC	the quadrant gives the sign: 3rd, $\cos$ negative	the sign gives the quadrants: $\cos$ negative, 2nd and 3rd
3. answer	$\cos 200^\circ = -\cos 20^\circ$	every angle in the range: $x = 126.9^\circ$ or $x = 233.1^\circ$

Step	Example: $\cos x = -0.6$ , $0^\circ \leq x \leq 360^\circ$
1. basic angle from the positive value	$\alpha = \cos^{-1} 0.6$ , so $\alpha = 53.1^\circ$
2. quadrants from ASTC	$\cos$ negative: 2nd and 3rd
3. every angle in the range	$x = 180^\circ - 53.1^\circ$ or $x = 180^\circ + 53.1^\circ$
	$x = 126.9^\circ$ or $x = 233.1^\circ$

Rearrange first: 2 sin x + 1 = 0

\displaystyle \sin x = -\dfrac{1}{2}

e.g.

\displaystyle \alpha = \sin^{-1} \dfrac{1}{2}

\displaystyle \alpha = 30^\circ

$\sin$ negative: 3rd and 4th quadrants
$x = 210^\circ$ or $x = 330^\circ$

In radians: sin x = 0.4, 0 ≤ x ≤ 2π

\displaystyle \sin x = 0.4

calculator in radian mode: $\alpha = \sin^{-1} 0.4$ , which is $0.4115$
$\sin$ positive: 1st and 2nd quadrants
$x = 0.412$ or $x = \pi - 0.4115$ , which is $2.73$ (3 s.f.)

📖 Worked example

Solve $\cos x = -0.6$ for $0^\circ \leq x \leq 360^\circ$ . Give your answers correct to 1 decimal place.

Step 1 of 3

Basic angle: use the positive value

\alpha = \cos^{-1}\left(0.6\right)

= 53.1^\circ

Quadrant	Degrees	Radians ( $\pi = 180^\circ$ )
1st	$\alpha$	$\alpha$
2nd	$180^\circ - \alpha$	$\pi - \alpha$
3rd	$180^\circ + \alpha$	$\pi + \alpha$
4th	$360^\circ - \alpha$	$2\pi - \alpha$

A range in radians: answers in radians

\displaystyle 2\sin x - 1 = 0,\ 0 \leq x \leq 2\pi

$\sin x = \dfrac{1}{2}$ , $\alpha = \dfrac{\pi}{6}$ ; $\sin$ positive: 1st and 2nd
$x = \dfrac{\pi}{6}$ or $x = \pi - \dfrac{\pi}{6}$
$x = \dfrac{5\pi}{6}$

📖 Worked example

Solve $2\sin x - 1 = 0$ for $0 \leq x \leq 2\pi$ , giving your answers in terms of $\pi$ .

Step 1 of 4

Rearrange to ratio = number

\sin x = + \dfrac{1}{2}

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3. Multiple angles and shifted angles

Multiple angle, 0° ≤ x ≤ 360°: let u = 2x

\displaystyle \sin 2x = \dfrac{1}{2}

range: $0^\circ \leq u \leq 720^\circ$
$u = 30^\circ$ or $u = 150^\circ$ or $u = 390^\circ$ or $u = 510^\circ$
$x = 15^\circ$ or $x = 75^\circ$ or $x = 195^\circ$ or $x = 255^\circ$

Shifted angle: let u = x − 30°

\displaystyle \cos(x - 30^\circ) = \dfrac{1}{2}

range: $-30^\circ \leq u \leq 330^\circ$
$u = 60^\circ$ or $u = 300^\circ$
$x = 90^\circ$ or $x = 330^\circ$

📖 Worked example

Solve $\sin 3x = -\dfrac{1}{2}$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 5

Let u = 3x: change the range first

\text{Lowest } u = 0^\circ

\text{Highest } u = 1080^\circ

📖 Worked example

Solve $\sin 2x = \dfrac{1}{2}$ for $0 \leq x \leq \pi$ , giving your answers in terms of $\pi$ .

Step 1 of 6

Let u = 2x: change the range first

\text{Lowest } u = 0

\text{Highest } u = 2\pi

Practise this →

4. Quadratics and reciprocal ratios

Name	Formula	Example
secant	$\sec x = \dfrac{1}{\cos x}$	$\sec x = 2$ , so $\cos x = \dfrac{1}{2}$
cosecant	$\operatorname{cosec} x = \dfrac{1}{\sin x}$	$\operatorname{cosec} x = 2$ , so $\sin x = \dfrac{1}{2}$
cotangent	$\cot x = \dfrac{1}{\tan x}$	$\cot x = 1$ , so $\tan x = 1$

Quadratic in sin x: let y = sin x

\displaystyle 2\sin^{2} x - \sin x - 1 = 0

e.g.

\displaystyle 2y^{2} - y - 1 = 0

\displaystyle (2y + 1)(y - 1) = 0

\displaystyle y = -\dfrac{1}{2} \text{ or } y = 1

$\sin x = -\dfrac{1}{2}$ : $x = 210^\circ$ or $x = 330^\circ$
$\sin x = 1$ : $x = 90^\circ$

📖 Worked example

Solve $2\sin^2 x - \sin x - 1 = 0$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 6

Let y = sin x: it's a quadratic in y

2y^2 - y - 1 = 0

📖 Worked example

Solve $\sec x = 2$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 4

sec x = 1 ÷ cos x: turn it round

\cos x = \dfrac{1}{2}

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5. Trigonometric models

$y = a\sin (bt)^\circ + c$	Formula	$d = 7 + 3\sin (30t)^\circ$
amplitude	$\|a\|$	$3$
period	$\dfrac{360}{b}$	$\dfrac{360}{30} = 12$ hours
centre line	$y = c$	$d = 7$
maximum	$c + \|a\|$	$7 + 3 = 10$
minimum	$c - \|a\|$	$7 - 3 = 4$

d = 7 + 3 sin(30t)° from t = 0 to 24 hours: centre line d = 7, greatest 10 at t = 3 and 15, least 4 at t = 9 and 21; the period of 12 hours and the amplitude 3 marked.

📖 Worked example

The depth of water, $d$ m, at the entrance to a harbour $t$ hours after midnight is given by $d = 7 + 3\sin (30t)^\circ$ . Find all the values of $t$ , for $0 \leq t \leq 24$ , for which $d = 8.5$ .

Step 1 of 6

Put d = 8.5 in and make sin the subject

7 + 3\sin (30t)^\circ = 8.5

3\sin (30t)^\circ = 8.5 - 7

\sin (30t)^\circ = + \dfrac{1}{2}

Practise this →

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Sec 3 · Additional Mathematics

Trigonometric equations

Practise trigonometric equations with step-by-step help. Free account.

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1. Principal values

Function	Principal value	In radians	Example
$\sin^{-1}$	$-90^\circ \leq \theta \leq 90^\circ$	$-\dfrac{\pi}{2} \leq \theta \leq \dfrac{\pi}{2}$	$\sin^{-1}\left(-\dfrac{1}{2}\right) = -30^\circ$
$\cos^{-1}$	$0^\circ \leq \theta \leq 180^\circ$	$0 \leq \theta \leq \pi$	$\cos^{-1}\left(-\dfrac{1}{2}\right) = 120^\circ$
$\tan^{-1}$	$-90^\circ < \theta < 90^\circ$	$-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}$	$\tan^{-1}(-1) = -45^\circ$

📖 Worked example

Without using a calculator, find the principal value, in degrees, of $\cos^{-1}\left(-\dfrac{1}{2}\right)$ .

Step 1 of 3

Find the basic angle (use the positive value)

\text{Basic angle} = 60^\circ

Practise this →

2. Solving sin x = k, cos x = k, tan x = k

ASTC + basic angle	Ch 9: angle → value	Ch 10: value → angles
given	$\cos 200^\circ$	$\cos x = -0.6$
1. basic angle $\alpha$	from the angle: $200^\circ - 180^\circ = 20^\circ$	from the positive value: $\cos^{-1} 0.6 = 53.1^\circ$
2. ASTC	the quadrant gives the sign: 3rd, $\cos$ negative	the sign gives the quadrants: $\cos$ negative, 2nd and 3rd
3. answer	$\cos 200^\circ = -\cos 20^\circ$	every angle in the range: $x = 126.9^\circ$ or $x = 233.1^\circ$

Step	Example: $\cos x = -0.6$ , $0^\circ \leq x \leq 360^\circ$
1. basic angle from the positive value	$\alpha = \cos^{-1} 0.6$ , so $\alpha = 53.1^\circ$
2. quadrants from ASTC	$\cos$ negative: 2nd and 3rd
3. every angle in the range	$x = 180^\circ - 53.1^\circ$ or $x = 180^\circ + 53.1^\circ$
	$x = 126.9^\circ$ or $x = 233.1^\circ$

Rearrange first: 2 sin x + 1 = 0

\displaystyle \sin x = -\dfrac{1}{2}

e.g.

\displaystyle \alpha = \sin^{-1} \dfrac{1}{2}

\displaystyle \alpha = 30^\circ

$\sin$ negative: 3rd and 4th quadrants
$x = 210^\circ$ or $x = 330^\circ$

In radians: sin x = 0.4, 0 ≤ x ≤ 2π

\displaystyle \sin x = 0.4

calculator in radian mode: $\alpha = \sin^{-1} 0.4$ , which is $0.4115$
$\sin$ positive: 1st and 2nd quadrants
$x = 0.412$ or $x = \pi - 0.4115$ , which is $2.73$ (3 s.f.)

📖 Worked example

Solve $\cos x = -0.6$ for $0^\circ \leq x \leq 360^\circ$ . Give your answers correct to 1 decimal place.

Step 1 of 3

Basic angle: use the positive value

\alpha = \cos^{-1}\left(0.6\right)

= 53.1^\circ

Quadrant	Degrees	Radians ( $\pi = 180^\circ$ )
1st	$\alpha$	$\alpha$
2nd	$180^\circ - \alpha$	$\pi - \alpha$
3rd	$180^\circ + \alpha$	$\pi + \alpha$
4th	$360^\circ - \alpha$	$2\pi - \alpha$

A range in radians: answers in radians

\displaystyle 2\sin x - 1 = 0,\ 0 \leq x \leq 2\pi

$\sin x = \dfrac{1}{2}$ , $\alpha = \dfrac{\pi}{6}$ ; $\sin$ positive: 1st and 2nd
$x = \dfrac{\pi}{6}$ or $x = \pi - \dfrac{\pi}{6}$
$x = \dfrac{5\pi}{6}$

📖 Worked example

Solve $2\sin x - 1 = 0$ for $0 \leq x \leq 2\pi$ , giving your answers in terms of $\pi$ .

Step 1 of 4

Rearrange to ratio = number

\sin x = + \dfrac{1}{2}

Practise this →

3. Multiple angles and shifted angles

Multiple angle, 0° ≤ x ≤ 360°: let u = 2x

\displaystyle \sin 2x = \dfrac{1}{2}

range: $0^\circ \leq u \leq 720^\circ$
$u = 30^\circ$ or $u = 150^\circ$ or $u = 390^\circ$ or $u = 510^\circ$
$x = 15^\circ$ or $x = 75^\circ$ or $x = 195^\circ$ or $x = 255^\circ$

Shifted angle: let u = x − 30°

\displaystyle \cos(x - 30^\circ) = \dfrac{1}{2}

range: $-30^\circ \leq u \leq 330^\circ$
$u = 60^\circ$ or $u = 300^\circ$
$x = 90^\circ$ or $x = 330^\circ$

📖 Worked example

Solve $\sin 3x = -\dfrac{1}{2}$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 5

Let u = 3x: change the range first

\text{Lowest } u = 0^\circ

\text{Highest } u = 1080^\circ

📖 Worked example

Solve $\sin 2x = \dfrac{1}{2}$ for $0 \leq x \leq \pi$ , giving your answers in terms of $\pi$ .

Step 1 of 6

Let u = 2x: change the range first

\text{Lowest } u = 0

\text{Highest } u = 2\pi

Practise this →

4. Quadratics and reciprocal ratios

Name	Formula	Example
secant	$\sec x = \dfrac{1}{\cos x}$	$\sec x = 2$ , so $\cos x = \dfrac{1}{2}$
cosecant	$\operatorname{cosec} x = \dfrac{1}{\sin x}$	$\operatorname{cosec} x = 2$ , so $\sin x = \dfrac{1}{2}$
cotangent	$\cot x = \dfrac{1}{\tan x}$	$\cot x = 1$ , so $\tan x = 1$

Quadratic in sin x: let y = sin x

\displaystyle 2\sin^{2} x - \sin x - 1 = 0

e.g.

\displaystyle 2y^{2} - y - 1 = 0

\displaystyle (2y + 1)(y - 1) = 0

\displaystyle y = -\dfrac{1}{2} \text{ or } y = 1

$\sin x = -\dfrac{1}{2}$ : $x = 210^\circ$ or $x = 330^\circ$
$\sin x = 1$ : $x = 90^\circ$

📖 Worked example

Solve $2\sin^2 x - \sin x - 1 = 0$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 6

Let y = sin x: it's a quadratic in y

2y^2 - y - 1 = 0

📖 Worked example

Solve $\sec x = 2$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 4

sec x = 1 ÷ cos x: turn it round

\cos x = \dfrac{1}{2}

Practise this →

5. Trigonometric models

$y = a\sin (bt)^\circ + c$	Formula	$d = 7 + 3\sin (30t)^\circ$
amplitude	$\|a\|$	$3$
period	$\dfrac{360}{b}$	$\dfrac{360}{30} = 12$ hours
centre line	$y = c$	$d = 7$
maximum	$c + \|a\|$	$7 + 3 = 10$
minimum	$c - \|a\|$	$7 - 3 = 4$

📖 Worked example

The depth of water, $d$ m, at the entrance to a harbour $t$ hours after midnight is given by $d = 7 + 3\sin (30t)^\circ$ . Find all the values of $t$ , for $0 \leq t \leq 24$ , for which $d = 8.5$ .

Step 1 of 6

Put d = 8.5 in and make sin the subject

7 + 3\sin (30t)^\circ = 8.5

3\sin (30t)^\circ = 8.5 - 7

\sin (30t)^\circ = + \dfrac{1}{2}

Practise this →

Try a question: Trigonometric equations

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