Sec 3 · Additional Mathematics

Trigonometric functions and graphs

Practise trigonometric functions and graphs with step-by-step help. Free account.

1. Exact values and radians

	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$ or $\dfrac{1}{\sqrt{2}}$	$\dfrac{\sqrt{3}}{2}$	$1$
$\cos$	$1$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$ or $\dfrac{1}{\sqrt{2}}$	$\dfrac{1}{2}$	$0$
$\tan$	$0$	$\dfrac{\sqrt{3}}{3}$ or $\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined
radians	$0$	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{2}$

Two right-angled triangles. One has angles 30° and 60° with sides √3 and 1 and hypotenuse 2. The other has two 45° angles with sides 1 and 1 and hypotenuse √2.

Radians and degrees

\displaystyle \pi \text{ rad} = 180^\circ

$\dfrac{\pi}{6} = \dfrac{180^\circ}{6}$ , which is $30^\circ$

Combine exact values

\displaystyle \sin 60^\circ \cos 30^\circ

e.g.

\displaystyle \dfrac{\sqrt{3}}{2} \times \dfrac{\sqrt{3}}{2}

\displaystyle = \dfrac{3}{4}

📖 Worked example

Without using a calculator, find the exact value of $\cos \dfrac{\pi}{6}$ .

Step 1 of 2

π radians = 180°: change to degrees first

\dfrac{\pi}{6} = \dfrac{180^\circ}{6}

= 30^\circ

Practise this →

2. Angles of any size

ASTC diagram: All positive in the first quadrant, Sine in the second, Tan in the third, Cos in the fourth. The angles α, 180° − α, 180° + α and 360° − α each make the basic angle α with the x-axis.

Quadrant	Angle	Positive
1st	$\alpha$	all
2nd	$180^\circ - \alpha$	$\sin$
3rd	$180^\circ + \alpha$	$\tan$
4th	$360^\circ - \alpha$	$\cos$

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 $x = 233.1^\circ$

2nd quadrant: sin is positive

\displaystyle \sin 150^\circ

e.g.

\displaystyle \sin 150^\circ

\displaystyle = \sin 30^\circ

\displaystyle = \dfrac{1}{2}

3rd quadrant: cos is negative

\displaystyle \cos 200^\circ

e.g.

\displaystyle \cos 200^\circ

\displaystyle = -\cos 20^\circ

4th quadrant: tan is negative

\displaystyle \tan 315^\circ

e.g.

\displaystyle \tan 315^\circ

\displaystyle = -\tan 45^\circ

\displaystyle = -1

📖 Worked example

Express $\cos 110^\circ$ in terms of a trigonometric ratio of its basic angle.

Step 1 of 3

Which quadrant is the angle in?

\text{Quadrant} = 2

Practise this →

3. Graphs of sine, cosine and tangent

y = sin x and y = cos x from 0° to 360°, each between −1 and 1 with period 360°. y = tan x with period 180° and asymptotes at 90° and 270°.

$y = a\sin bx + c$ or $y = a\cos bx + c$	Formula	$y = 3\sin 2x + 1$
amplitude	$\|a\|$	$3$
period ( $b > 0$ )	$\dfrac{360^\circ}{b}$ or $\dfrac{2\pi}{b}$	$180^\circ$
maximum	$c + \|a\|$	$4$
minimum	$c - \|a\|$	$-2$
$y = a\tan bx$ : period	$\dfrac{180^\circ}{b}$ or $\dfrac{\pi}{b}$

y = 3 sin 2x + 1 from 0° to 360°: centre line y = 1, maximum 4 at 45° and 225°, minimum −2 at 135° and 315°; amplitude 3 and period 180° marked.

📖 Worked example

State the maximum value of $y = \sin 4x - 1$ .

Step 1 of 2

sin goes between −1 and 1

\text{Amplitude} = 1

Practise this →

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Sec 3 · Additional Mathematics

Trigonometric functions and graphs

Practise trigonometric functions and graphs with step-by-step help. Free account.

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1. Exact values and radians

	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$ or $\dfrac{1}{\sqrt{2}}$	$\dfrac{\sqrt{3}}{2}$	$1$
$\cos$	$1$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$ or $\dfrac{1}{\sqrt{2}}$	$\dfrac{1}{2}$	$0$
$\tan$	$0$	$\dfrac{\sqrt{3}}{3}$ or $\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined
radians	$0$	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{2}$

Radians and degrees

\displaystyle \pi \text{ rad} = 180^\circ

$\dfrac{\pi}{6} = \dfrac{180^\circ}{6}$ , which is $30^\circ$

Combine exact values

\displaystyle \sin 60^\circ \cos 30^\circ

e.g.

\displaystyle \dfrac{\sqrt{3}}{2} \times \dfrac{\sqrt{3}}{2}

\displaystyle = \dfrac{3}{4}

📖 Worked example

Without using a calculator, find the exact value of $\cos \dfrac{\pi}{6}$ .

Step 1 of 2

π radians = 180°: change to degrees first

\dfrac{\pi}{6} = \dfrac{180^\circ}{6}

= 30^\circ

Practise this →

2. Angles of any size

Quadrant	Angle	Positive
1st	$\alpha$	all
2nd	$180^\circ - \alpha$	$\sin$
3rd	$180^\circ + \alpha$	$\tan$
4th	$360^\circ - \alpha$	$\cos$

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 $x = 233.1^\circ$

2nd quadrant: sin is positive

\displaystyle \sin 150^\circ

e.g.

\displaystyle \sin 150^\circ

\displaystyle = \sin 30^\circ

\displaystyle = \dfrac{1}{2}

3rd quadrant: cos is negative

\displaystyle \cos 200^\circ

e.g.

\displaystyle \cos 200^\circ

\displaystyle = -\cos 20^\circ

4th quadrant: tan is negative

\displaystyle \tan 315^\circ

e.g.

\displaystyle \tan 315^\circ

\displaystyle = -\tan 45^\circ

\displaystyle = -1

📖 Worked example

Express $\cos 110^\circ$ in terms of a trigonometric ratio of its basic angle.

Step 1 of 3

Which quadrant is the angle in?

\text{Quadrant} = 2

Practise this →

3. Graphs of sine, cosine and tangent

$y = a\sin bx + c$ or $y = a\cos bx + c$	Formula	$y = 3\sin 2x + 1$
amplitude	$\|a\|$	$3$
period ( $b > 0$ )	$\dfrac{360^\circ}{b}$ or $\dfrac{2\pi}{b}$	$180^\circ$
maximum	$c + \|a\|$	$4$
minimum	$c - \|a\|$	$-2$
$y = a\tan bx$ : period	$\dfrac{180^\circ}{b}$ or $\dfrac{\pi}{b}$

📖 Worked example

State the maximum value of $y = \sin 4x - 1$ .

Step 1 of 2

sin goes between −1 and 1

\text{Amplitude} = 1

Practise this →

Try a question: Trigonometric functions and graphs

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise trigonometric functions and graphs with step-by-step help

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