Sec 4 · Additional Mathematics

Trigonometric identities and formulae

Practise trigonometric identities and formulae with step-by-step help. Free account.

1. Basic identities and reciprocal ratios

Identity	Example
$\tan A = \dfrac{\sin A}{\cos A}$	$\tan 45^\circ = \dfrac{\sin 45^\circ}{\cos 45^\circ}$
$\sec A = \dfrac{1}{\cos A}$	$\sec 60^\circ = \dfrac{1}{\cos 60^\circ}$ , which is $2$
$\operatorname{cosec} A = \dfrac{1}{\sin A}$	$\operatorname{cosec} 30^\circ = \dfrac{1}{\sin 30^\circ}$ , which is $2$
$\cot A = \dfrac{1}{\tan A}$ , which is $\dfrac{\cos A}{\sin A}$	$\cot 45^\circ = \dfrac{1}{\tan 45^\circ}$ , which is $1$
$\sin^{2} A + \cos^{2} A = 1$	$\sin^{2} 30^\circ + \cos^{2} 30^\circ = 1$
$1 + \tan^{2} A = \sec^{2} A$	$1 + \tan^{2} 45^\circ = \sec^{2} 45^\circ$
$1 + \cot^{2} A = \operatorname{cosec}^{2} A$	$1 + \cot^{2} 45^\circ = \operatorname{cosec}^{2} 45^\circ$

Quadrant	Angle	Positive (ASTC)
1	$0^\circ$ to $90^\circ$	All
2	$90^\circ$ to $180^\circ$	$\sin$ (and $\operatorname{cosec}$ )
3	$180^\circ$ to $270^\circ$	$\tan$ (and $\cot$ )
4	$270^\circ$ to $360^\circ$	$\cos$ (and $\sec$ )

A obtuse: cos A from sin²A + cos²A = 1

\displaystyle \sin A = \dfrac{3}{5}

e.g.

\displaystyle \cos^{2} A = 1 - \left(\dfrac{3}{5}\right)^{2}

\displaystyle \cos^{2} A = \dfrac{16}{25}

$A$ obtuse: $\cos A$ is negative, so $\cos A = -\dfrac{4}{5}$

📖 Worked example

Given that $\sin A = \dfrac{3}{5}$ and $A$ is acute, find the exact value of $\cos A$ .

Step 1 of 3

Draw a right-angled triangle with the sizes in sin A (ignore the sign), and find the adjacent by Pythagoras' theorem

\text{Adjacent} = \sqrt{5^{2} - 3^{2}}

= 4

Practise this →

2. Addition formulae

Formula
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Exact value: 75° = 45° + 30°

\displaystyle \sin 75^\circ

e.g.

\displaystyle \sin(45^\circ + 30^\circ)

\displaystyle = \sin 45^\circ\cos 30^\circ + \cos 45^\circ\sin 30^\circ

\displaystyle = \dfrac{\sqrt{2}}{2} \times \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \times \dfrac{1}{2}

\displaystyle = \dfrac{\sqrt{6} + \sqrt{2}}{4}

📖 Worked example

Solve $\sin(x + 30^\circ) = 2\cos x$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 5

Expand the left side with the addition formula

\sin(x + 30^\circ) = \sin x\cos 30^\circ + \cos x\sin 30^\circ

\cos 30^\circ = \dfrac{\sqrt{3}}{2}

\sin 30^\circ = \dfrac{1}{2}

In radians	As a sum or difference	In degrees
$\dfrac{\pi}{12}$	$\dfrac{\pi}{4} - \dfrac{\pi}{6}$	$45^\circ - 30^\circ$
$\dfrac{5\pi}{12}$	$\dfrac{\pi}{4} + \dfrac{\pi}{6}$	$45^\circ + 30^\circ$
$\dfrac{7\pi}{12}$	$\dfrac{\pi}{3} + \dfrac{\pi}{4}$	$60^\circ + 45^\circ$

sec, cosec, cot: take the reciprocal, then rationalise

\displaystyle \sec 15^\circ

$\cos 15^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ , so $\sec 15^\circ = \dfrac{4}{\sqrt{6} + \sqrt{2}}$
$= \dfrac{4(\sqrt{6} - \sqrt{2})}{6 - 2}$
$= \sqrt{6} - \sqrt{2}$

📖 Worked example

Without using a calculator, find the exact value of $\cos \dfrac{\pi}{12}$ .

Step 1 of 4

Write the angle as a sum or difference of angles whose exact values you know

\dfrac{\pi}{12} = \dfrac{\pi}{4} - \dfrac{\pi}{6}

📖 Worked example

Without using a calculator, find the exact value of $\sec 15^\circ$ , in simplest surd form with a rational denominator.

Step 1 of 3

First find the exact value of the ratio it is the reciprocal of, with an addition formula

\cos 15^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}

Practise this →

3. Double angle formulae

Formula	Use when
$\sin 2A = 2\sin A\cos A$
$\cos 2A = \cos^{2} A - \sin^{2} A$
$\cos 2A = 2\cos^{2} A - 1$	only $\cos$ is wanted
$\cos 2A = 1 - 2\sin^{2} A$	only $\sin$ is wanted
$\tan 2A = \dfrac{2\tan A}{1 - \tan^{2} A}$

tan 2A, with A acute

\displaystyle \tan A = \dfrac{5}{12}

e.g.

\displaystyle \dfrac{2 \times \dfrac{5}{12}}{1 - \left(\dfrac{5}{12}\right)^{2}}

\displaystyle = \dfrac{120}{119}

📖 Worked example

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

Step 1 of 7

Replace cos 2x by the form that leaves only the ratio on the right-hand side

2\cos^{2} x - 1 = \cos x

Practise this →

4. The R-formula

Name	Formula	Example: $3\sin\theta + 4\cos\theta$
form	$a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$	$5\sin(\theta + 53.13^\circ)$
$R$	$\sqrt{a^{2} + b^{2}}$	$\sqrt{3^{2} + 4^{2}}$ , so $5$
$\alpha$	$\tan^{-1} \dfrac{b}{a}$	$\tan^{-1} \dfrac{4}{3}$ , so $53.13^\circ$
maximum	$R$	$5$
minimum	$-R$	$-5$

Expression	Write as
$a\sin\theta - b\cos\theta$	$R\sin(\theta - \alpha)$
$a\cos\theta + b\sin\theta$	$R\cos(\theta - \alpha)$
$a\cos\theta - b\sin\theta$	$R\cos(\theta + \alpha)$

📖 Worked example

Express $2\sin\theta + 3\cos\theta$ in the form $R\sin(\theta + \alpha)$ , where $R > 0$ and $0^\circ < \alpha < 90^\circ$ . Hence solve $2\sin\theta + 3\cos\theta = -1$ for $0^\circ \leq \theta \leq 360^\circ$ .

Step 1 of 9

Expand the R form with the addition formula, and compare coefficients

R\sin(\theta + \alpha) = R\sin\theta\cos\alpha + R\cos\theta\sin\alpha

R\cos\alpha = 2

R\sin\alpha = 3

Practise this →

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Sec 4 · Additional Mathematics

Trigonometric identities and formulae

Practise trigonometric identities and formulae with step-by-step help. Free account.

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1. Basic identities and reciprocal ratios

Identity	Example
$\tan A = \dfrac{\sin A}{\cos A}$	$\tan 45^\circ = \dfrac{\sin 45^\circ}{\cos 45^\circ}$
$\sec A = \dfrac{1}{\cos A}$	$\sec 60^\circ = \dfrac{1}{\cos 60^\circ}$ , which is $2$
$\operatorname{cosec} A = \dfrac{1}{\sin A}$	$\operatorname{cosec} 30^\circ = \dfrac{1}{\sin 30^\circ}$ , which is $2$
$\cot A = \dfrac{1}{\tan A}$ , which is $\dfrac{\cos A}{\sin A}$	$\cot 45^\circ = \dfrac{1}{\tan 45^\circ}$ , which is $1$
$\sin^{2} A + \cos^{2} A = 1$	$\sin^{2} 30^\circ + \cos^{2} 30^\circ = 1$
$1 + \tan^{2} A = \sec^{2} A$	$1 + \tan^{2} 45^\circ = \sec^{2} 45^\circ$
$1 + \cot^{2} A = \operatorname{cosec}^{2} A$	$1 + \cot^{2} 45^\circ = \operatorname{cosec}^{2} 45^\circ$

Quadrant	Angle	Positive (ASTC)
1	$0^\circ$ to $90^\circ$	All
2	$90^\circ$ to $180^\circ$	$\sin$ (and $\operatorname{cosec}$ )
3	$180^\circ$ to $270^\circ$	$\tan$ (and $\cot$ )
4	$270^\circ$ to $360^\circ$	$\cos$ (and $\sec$ )

A obtuse: cos A from sin²A + cos²A = 1

\displaystyle \sin A = \dfrac{3}{5}

e.g.

\displaystyle \cos^{2} A = 1 - \left(\dfrac{3}{5}\right)^{2}

\displaystyle \cos^{2} A = \dfrac{16}{25}

$A$ obtuse: $\cos A$ is negative, so $\cos A = -\dfrac{4}{5}$

📖 Worked example

Given that $\sin A = \dfrac{3}{5}$ and $A$ is acute, find the exact value of $\cos A$ .

Step 1 of 3

Draw a right-angled triangle with the sizes in sin A (ignore the sign), and find the adjacent by Pythagoras' theorem

\text{Adjacent} = \sqrt{5^{2} - 3^{2}}

= 4

Practise this →

2. Addition formulae

Formula
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Exact value: 75° = 45° + 30°

\displaystyle \sin 75^\circ

e.g.

\displaystyle \sin(45^\circ + 30^\circ)

\displaystyle = \sin 45^\circ\cos 30^\circ + \cos 45^\circ\sin 30^\circ

\displaystyle = \dfrac{\sqrt{2}}{2} \times \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \times \dfrac{1}{2}

\displaystyle = \dfrac{\sqrt{6} + \sqrt{2}}{4}

📖 Worked example

Solve $\sin(x + 30^\circ) = 2\cos x$ for $0^\circ \leq x \leq 360^\circ$ .

Step 1 of 5

Expand the left side with the addition formula

\sin(x + 30^\circ) = \sin x\cos 30^\circ + \cos x\sin 30^\circ

\cos 30^\circ = \dfrac{\sqrt{3}}{2}

\sin 30^\circ = \dfrac{1}{2}

In radians	As a sum or difference	In degrees
$\dfrac{\pi}{12}$	$\dfrac{\pi}{4} - \dfrac{\pi}{6}$	$45^\circ - 30^\circ$
$\dfrac{5\pi}{12}$	$\dfrac{\pi}{4} + \dfrac{\pi}{6}$	$45^\circ + 30^\circ$
$\dfrac{7\pi}{12}$	$\dfrac{\pi}{3} + \dfrac{\pi}{4}$	$60^\circ + 45^\circ$

sec, cosec, cot: take the reciprocal, then rationalise

\displaystyle \sec 15^\circ

$\cos 15^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ , so $\sec 15^\circ = \dfrac{4}{\sqrt{6} + \sqrt{2}}$
$= \dfrac{4(\sqrt{6} - \sqrt{2})}{6 - 2}$
$= \sqrt{6} - \sqrt{2}$

📖 Worked example

Without using a calculator, find the exact value of $\cos \dfrac{\pi}{12}$ .

Step 1 of 4

Write the angle as a sum or difference of angles whose exact values you know

\dfrac{\pi}{12} = \dfrac{\pi}{4} - \dfrac{\pi}{6}

📖 Worked example

Without using a calculator, find the exact value of $\sec 15^\circ$ , in simplest surd form with a rational denominator.

Step 1 of 3

First find the exact value of the ratio it is the reciprocal of, with an addition formula

\cos 15^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}

Practise this →

3. Double angle formulae

Formula	Use when
$\sin 2A = 2\sin A\cos A$
$\cos 2A = \cos^{2} A - \sin^{2} A$
$\cos 2A = 2\cos^{2} A - 1$	only $\cos$ is wanted
$\cos 2A = 1 - 2\sin^{2} A$	only $\sin$ is wanted
$\tan 2A = \dfrac{2\tan A}{1 - \tan^{2} A}$

tan 2A, with A acute

\displaystyle \tan A = \dfrac{5}{12}

e.g.

\displaystyle \dfrac{2 \times \dfrac{5}{12}}{1 - \left(\dfrac{5}{12}\right)^{2}}

\displaystyle = \dfrac{120}{119}

📖 Worked example

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

Step 1 of 7

Replace cos 2x by the form that leaves only the ratio on the right-hand side

2\cos^{2} x - 1 = \cos x

Practise this →

4. The R-formula

Name	Formula	Example: $3\sin\theta + 4\cos\theta$
form	$a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$	$5\sin(\theta + 53.13^\circ)$
$R$	$\sqrt{a^{2} + b^{2}}$	$\sqrt{3^{2} + 4^{2}}$ , so $5$
$\alpha$	$\tan^{-1} \dfrac{b}{a}$	$\tan^{-1} \dfrac{4}{3}$ , so $53.13^\circ$
maximum	$R$	$5$
minimum	$-R$	$-5$

Expression	Write as
$a\sin\theta - b\cos\theta$	$R\sin(\theta - \alpha)$
$a\cos\theta + b\sin\theta$	$R\cos(\theta - \alpha)$
$a\cos\theta - b\sin\theta$	$R\cos(\theta + \alpha)$

📖 Worked example

Step 1 of 9

Expand the R form with the addition formula, and compare coefficients

R\sin(\theta + \alpha) = R\sin\theta\cos\alpha + R\cos\theta\sin\alpha

R\cos\alpha = 2

R\sin\alpha = 3

Practise this →

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