Cambridge · IGCSE Maths 0580

Cambridge IGCSE Maths: Trigonometry — Practice Questions & Answers

Q: In a right-angled triangle the two shorter sides are $3$ and $4$. What is the length of the hypotenuse?

$5$. By Pythagoras, $h=\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5$.

Q: Which ratio equals $\sin\theta$ in a right-angled triangle?

$\frac{\text{opposite}}{\text{hypotenuse}}$. SOHCAHTOA: sine equals opposite divided by hypotenuse.

Q: What is the exact value of $\cos 60^{\circ}$?

$\frac{1}{2}$. $\cos 60^{\circ}=\frac{1}{2}$ is a standard exact value.

Q: What is the exact value of $\tan 45^{\circ}$?

$1$. At $45^{\circ}$ the opposite and adjacent are equal, so $\tan 45^{\circ}=1$.

Q: A ladder leans against a wall. The angle measured from the ground up to the ladder is an example of which angle?

Angle of elevation. An angle measured upward from the horizontal is an angle of elevation.

Q: A right-angled triangle has hypotenuse $13$ and one shorter side $5$. The other shorter side is:

$12$. $\sqrt{13^{2}-5^{2}}=\sqrt{169-25}=\sqrt{144}=12$.

Q: A bearing is always measured:

Clockwise from north as a three-figure angle. Bearings are measured clockwise from north and written with three figures.

Q: In triangle $ABC$, the area formula $\frac{1}{2}ab\sin C$ uses the angle $C$ that lies:

Between sides $a$ and $b$. The angle used must be the included angle between the two sides $a$ and $b$.

Q: A right-angled triangle has legs of length $6$ and $8$. The length of the hypotenuse is _____.

10. $\sqrt{6^{2}+8^{2}}=\sqrt{36+64}=\sqrt{100}=10$.

Q: In a right-angled triangle the opposite side to angle $\theta$ is $7$ and the hypotenuse is $10$. To one decimal place, $\theta$ is:

$44.4^{\circ}$. $\sin\theta=\frac{7}{10}=0.7$, so $\theta=\sin^{-1}(0.7)\approx 44.4^{\circ}$.

Trigonometry links the sides and angles of triangles. This guide covers Pythagoras' theorem, the right-angled ratios sine, cosine and tangent (SOHCAHTOA), exact values, angles of elevation and depression, bearings, the sine and cosine rules for any triangle, the area formula, three-dimensional problems, and the shapes of the sin, cos and tan graphs.

258 practice questions available for this topic — here are 10 with full answers and explanations.

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Practice questions with answers

1 Multiple choice · Easy

In a right-angled triangle the two shorter sides are $3$ and $4$. What is the length of the hypotenuse?

$5$
$7$
$\sqrt{7}$
$25$

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2 Multiple choice · Easy

Which ratio equals $\sin\theta$ in a right-angled triangle?

$\frac{\text{opposite}}{\text{hypotenuse}}$
$\frac{\text{adjacent}}{\text{hypotenuse}}$
$\frac{\text{opposite}}{\text{adjacent}}$
$\frac{\text{hypotenuse}}{\text{opposite}}$

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3 Multiple choice · Easy

What is the exact value of $\cos 60^{\circ}$?

$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
$\frac{\sqrt{2}}{2}$
$1$

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4 Multiple choice · Easy

What is the exact value of $\tan 45^{\circ}$?

$1$
$0$
$\frac{1}{2}$
$\sqrt{3}$

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5 Multiple choice · Easy

A ladder leans against a wall. The angle measured from the ground up to the ladder is an example of which angle?

Angle of elevation
Angle of depression
Bearing
Reflex angle

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6 Multiple choice · Easy

A right-angled triangle has hypotenuse $13$ and one shorter side $5$. The other shorter side is:

$12$
$8$
$\sqrt{194}$
$18$

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7 Multiple choice · Easy

A bearing is always measured:

Clockwise from north as a three-figure angle
Anticlockwise from north
Clockwise from east
From the horizontal ground

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8 Multiple choice · Easy

In triangle $ABC$, the area formula $\frac{1}{2}ab\sin C$ uses the angle $C$ that lies:

Between sides $a$ and $b$
Opposite side $a$
Opposite side $b$
At the right angle only

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9 Fill in the blank · Easy

A right-angled triangle has legs of length $6$ and $8$. The length of the hypotenuse is .

10 Multiple choice · Medium

In a right-angled triangle the opposite side to angle $\theta$ is $7$ and the hypotenuse is $10$. To one decimal place, $\theta$ is:

$44.4^{\circ}$
$45.6^{\circ}$
$35.0^{\circ}$
$0.7^{\circ}$

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Key terms in Trigonometry

Trigonometry: The study of relationships between angles and sides of triangles.

Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

Opposite side: The side of a right-angled triangle facing a chosen angle.

Adjacent side: The side of a right-angled triangle next to a chosen angle and the right angle.

Sine: The ratio of the opposite side to the hypotenuse in a right-angled triangle.

Cosine: The ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Tangent: The ratio of the opposite side to the adjacent side in a right-angled triangle.

Trigonometric ratio: A ratio of two sides of a right-angled triangle relating to an angle.

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Cambridge IGCSE Maths: Trigonometry — Practice Questions & Answers

Practice questions with answers

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Key terms in Trigonometry

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