Year 10 and 10A Trigonometry: free practice test with answers

Sample trigonometry questions with answers

Try each one before you open the answer.

1. Trigonometric ratios

   In triangle $PQR$, $\angle Q = 90^\circ$. The side opposite angle $\theta$ at $P$ is $QR = 5$ cm, the side adjacent to $\theta$ is $PQ = 12$ cm and the hypotenuse is $PR = 13$ cm.

   Write, as fractions:

   (a) $\cos \theta$

   (b) $\tan \theta$

   Show the answer

   Key idea: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ and $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.

   1. $\cos \theta = \frac{12}{13}$
   2. $\tan \theta = \frac{5}{12}$

   Answer: (a) $\frac{12}{13}$ (b) $\frac{5}{12}$

2. Finding unknown angles

   Use a calculator. In a right-angled triangle, the side opposite angle $\theta$ is 7 cm and the hypotenuse is 12 cm. Find $\theta$, correct to one decimal place.

   Show the answer

   Key idea: Opposite and hypotenuse are known, so use sine, then the inverse sine function.

   1. $\sin\theta = \frac{7}{12}$
   2. $\theta = \sin^{-1}\left(\frac{7}{12}\right) = 35.685\ldots^\circ \approx 35.7^\circ$

   Answer: $\theta \approx 35.7^\circ$

3. Applications in two dimensions

   A lifeguard sits in a tower 8 m above the ground and looks down at a swimmer on the beach. The swimmer is 15 m from the base of the tower along flat ground. Find the angle of depression from the lifeguard to the swimmer, correct to one decimal place.

   Show the answer

   Key idea: The angle of depression equals the angle inside the right-angled triangle at the top (alternate angles with the horizontal). Use the tangent ratio because you have the opposite side (8 m) and the adjacent side (15 m).

   1. Label the triangle: opposite $= 8$ m, adjacent $= 15$ m.
   2. Write the trig equation:\

   $$\tan(\theta) = \frac{8}{15}$$

   3. Solve for $\theta$:\

   $$\theta = \tan^{-1}\!\left(\frac{8}{15}\right)$$

   4. Calculate:\

   $$\theta \approx 28.1°$$

   Answer: The angle of depression is approximately $28.1°$.

4. Applications in three dimensions

   A rectangular prism has length $8$ m, width $6$ m and height $5$ m. Find the angle that the space diagonal makes with the base of the prism, correct to one decimal place.

   Show the answer

   Key idea: First find the diagonal of the base, then use trigonometry in the vertical right-angled triangle formed by that base diagonal and the space diagonal.

   1. Find the base diagonal $b$ using Pythagoras' theorem:\

   $$b = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ m}$$

   2. The space diagonal, the base diagonal and the height form a right-angled triangle. The side opposite the required angle $\theta$ is the height ($5$ m) and the side adjacent is the base diagonal ($10$ m).
   3. Apply the tangent ratio:\

   $$\tan\theta = \frac{5}{10} = 0.5$$

   4. Solve for $\theta$:\

   $$\theta = \tan^{-1}(0.5) \approx 26.6°$$

   Answer: The space diagonal makes an angle of approximately $26.6°$ with the base.

5. The sine rule

   In triangle $ABC$, angle $A = 47°$, angle $B = 63°$, and side $a = 14$ cm. Find side $b$, correct to one decimal place.

   Show the answer

   Key idea: When two angles and one opposite side are known, use the sine rule $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$.

   1. Write the sine rule with the known values:\

   $$\frac{14}{\sin 47°} = \frac{b}{\sin 63°}$$

   2. Solve for $b$:\

   $$b = \frac{14 \times \sin 63°}{\sin 47°}$$

   3. Evaluate:\

   $$b = \frac{14 \times 0.8910}{0.7314} \approx \frac{12.474}{0.7314} \approx 17.1 \text{ cm}$$

   Answer: $b \approx 17.1$ cm

6. The cosine rule

   In triangle $PQR$, $p = 9$ cm, $q = 11$ cm, and $r = 7$ cm. Find angle $R$, correct to the nearest degree.

   Show the answer

   Key idea: When all three sides are known, use the cosine rule in the form $\cos R = \dfrac{p^2 + q^2 - r^2}{2pq}$.

   1. Substitute the known values:\

   $$\cos R = \frac{9^2 + 11^2 - 7^2}{2 \times 9 \times 11}$$

   2. Simplify the numerator and denominator:\

   $$\cos R = \frac{81 + 121 - 49}{198} = \frac{153}{198}$$

   3. Divide:\

   $$\cos R \approx 0.7727$$

   4. Find angle $R$:\

   $$R = \cos^{-1}(0.7727) \approx 39.4° \approx 39°$$

   Answer: $R \approx 39°$

7. The unit circle

   Use the unit circle to find the exact value of:

   (a) $\cos 180^\circ$

   (b) $\sin 270^\circ$

   (c) $\cos 300^\circ$

   Show the answer

   Key idea: The point at angle $\theta$ on the unit circle is $(\cos \theta, \sin \theta)$; use reference angles for non-axis angles.

   1. At $180^\circ$ the point is $(-1, 0)$, so $\cos 180^\circ = -1$
   2. At $270^\circ$ the point is $(0, -1)$, so $\sin 270^\circ = -1$
   3. $300^\circ$ is in the fourth quadrant with reference angle $60^\circ$, and cosine is positive there, so $\cos 300^\circ = \cos 60^\circ = \frac{1}{2}$

   Answer: (a) $-1$ (b) $-1$ (c) $\frac{1}{2}$

8. Graphs of trigonometric functions

   For the graph of $y = 4\sin x$:

   (a) State the amplitude.

   (b) State the period, in degrees.

   (c) State the minimum value of $y$.

   Show the answer

   Key idea: For $y = a\sin x$, the amplitude is $|a|$ and the period is still $360^\circ$.

   1. Amplitude $= 4$
   2. Period $= 360^\circ$ (the $x$ is not multiplied by anything)
   3. The graph goes from $-4$ to $4$, so the minimum is $-4$

   Answer: (a) $4$ (b) $360^\circ$ (c) $-4$

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 10 trigonometry worksheet
• Year 10A pythagoras and trigonometry in 3d worksheet
• Year 10A sine rule, cosine rule and area worksheet
• Year 10A solving trigonometric equations worksheet