Trigonometry in right-angled triangles – Year 9 maths worksheet with answers to download as PDF

In a right-angled triangle the hypotenuse is opposite the right angle. The opposite side is across from the marked angle, and the adjacent side is next to it (not the hypotenuse).

1. Triangle \(ABC\) has a right angle at \(B\). For angle \(A\), name the hypotenuse, the opposite side and the adjacent side.
2. For angle \(C\) in the same triangle, name the opposite and adjacent sides.
3. Triangle \(PQR\) has a right angle at \(Q\). For angle \(P\), which side is opposite?
4. Which side is the hypotenuse of triangle \(PQR\)?

SOH CAH TOA: \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\). A right-angled triangle has sides 3, 4 and 5. The angle \(\theta\) is opposite the side 3.

1. Write \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\) as fractions.
2. Write the same three ratios for the other acute angle \(\phi\).
3. What do you notice about \(\sin\theta\) and \(\cos\phi\)?
4. A triangle has sides 5, 12, 13. Write \(\sin\), \(\cos\) and \(\tan\) of the angle opposite 5.

Make sure your calculator is in degree mode. Give each answer to 3 decimal places.

1. \(\sin30^\circ\) and \(\cos60^\circ\)
2. \(\tan45^\circ\)
3. \(\sin40^\circ\) and \(\cos25^\circ\)
4. \(\tan70^\circ\)

The hypotenuse and an angle are known and you need the opposite side, so use \(\text{opp}=\text{hyp}\times\sin\theta\). Give answers to 2 decimal places.

1. Hypotenuse 10 cm, angle \(30^\circ\).
2. Hypotenuse 8 cm, angle \(40^\circ\).
3. Hypotenuse 15 cm, angle \(55^\circ\).
4. Hypotenuse 20 cm, angle \(25^\circ\).

Decide which ratio to use, write the equation and find the length \(x\). Give answers to 2 decimal places.

1. Adjacent 6 cm, angle \(35^\circ\). Find the opposite side \(x\).
2. Adjacent 8 cm, angle \(50^\circ\). Find the hypotenuse \(x\).
3. Opposite 7 cm, angle \(40^\circ\). Find the adjacent side \(x\).
4. Adjacent \(x\) is needed from hypotenuse 12 cm and angle \(62^\circ\).

Use the inverse functions \(\sin^{-1}\), \(\cos^{-1}\) and \(\tan^{-1}\) to find the angle. Give answers to 1 decimal place.

1. \(\sin\theta=0.5\)
2. \(\cos\theta=0.8\)
3. \(\tan\theta=1.5\)
4. A right-angled triangle has opposite 6 cm and hypotenuse 10 cm. Find \(\theta\).

A 5 m ladder leans against a vertical wall. The foot of the ladder is 1.5 m from the wall.

1. Find the angle between the ladder and the ground, to 1 decimal place.
2. Find how high up the wall the ladder reaches, using sine.
3. Check your answer to (b) using Pythagoras' theorem.
4. Safety guidance says the angle should be between \(70^\circ\) and \(80^\circ\). Is the ladder placed safely?

Layla stands 20 m from the base of a tree. She measures the angle of elevation of the top of the tree as \(35^\circ\). Her eyes are 1.6 m above the ground.

1. Find the height \(h\) of the top of the tree above her eye level.
2. Find the height of the tree above the ground.
3. She moves back until the angle is \(25^\circ\). How far from the tree is she now?
4. Why is it important to add the height of her eyes?

A ramp is 4 m long and rises by 0.5 m.

1. Find the angle of the ramp to the horizontal.
2. Find the horizontal distance covered by the ramp.
3. The rule says the angle must not be more than \(5^\circ\). Is it allowed?
4. What height would a ramp of the same length have at \(5^\circ\)?

Rhys flies a kite on a 60 m string. The string makes an angle of \(50^\circ\) with the horizontal ground. (Ignore the sag in the string.)

1. Find the height of the kite above the ground, to the nearest metre.
2. Find how far along the ground the kite is from Rhys.
3. The angle later changes to \(65^\circ\) with the string length the same. Find the new height.
4. Which is higher, and by how much?

A playground slide is 3 m long and makes an angle of \(35^\circ\) with the ground.

1. Find the height of the top of the slide.
2. Find the horizontal distance from the foot of the slide to the point below the top.
3. Check using Pythagoras.
4. A second slide is 3 m long but only 1.2 m high. Find its angle.

A plane descends at an angle of \(3^\circ\) to the horizontal while it travels a horizontal distance of 30 km towards the airport.

1. Find the height lost, to 2 decimal places.
2. Find the distance flown through the air.
3. The plane starts at 10 km high. How far must it travel horizontally to reach the ground at \(3^\circ\)?
4. Why are descents shallow in practice?

Ben is 1.8 m tall and his shadow on level ground is 2.5 m long.

1. Find the angle of elevation of the sun, to 1 decimal place.
2. The shadow of a tower at the same moment is 40 m. Find the height of the tower.
3. Later the elevation is \(50^\circ\). How long is Ben's shadow?
4. Is the shadow longer or shorter when the sun is higher?

A cable car climbs at \(28^\circ\) to the horizontal up to a mountain station 600 m above the start.

1. Find the length of the cable.
2. Find the horizontal distance between the stations.
3. The cable car travels at 4 m/s. How long is the journey, in minutes and seconds?
4. A steeper cable at \(35^\circ\) to the same height is built. Find its length.

The table shows three right-angled triangles. The hypotenuse is \(h\), the opposite side is \(o\) and the adjacent side is \(a\) for the angle \(\theta\). All lengths are in centimetres.

1. Complete the table. Give answers to 1 decimal place.
2. Check triangle 1 using Pythagoras.
3. Show that triangle 3 is a 3, 4, 5 triangle enlarged. By what scale factor?
4. Find the other acute angle in triangle 3.

An isosceles triangle has two equal sides of 10 cm and an angle of \(40^\circ\) between them. The line from the apex to the midpoint of the base splits it into two right-angled triangles.

1. Find the angle at the apex of each right-angled triangle.
2. Find the half-base and the height.
3. Find the base of the isosceles triangle.
4. Find the area of the isosceles triangle.
5. Find the perimeter.

Anna stands 30 m from the foot of a building. She measures the angle of elevation to the top of the building as \(52^\circ\) and to the top of a flagpole on the roof as \(57^\circ\).

1. Find the height of the building.
2. Find the height of the top of the flagpole above the ground.
3. Find the height of the flagpole.
4. The flagpole is meant to be 8 m tall. Is the error in the measurement reasonable?
5. Anna's eyes are 1.5 m above the ground. Give the true height of the building.

From the top of a lighthouse 40 m above sea level, the angle of depression of a boat is \(12^\circ\).

1. Explain why the angle between the line of sight and the vertical is not the angle of depression.
2. Find the horizontal distance from the lighthouse to the boat.
3. A second boat in the same direction has an angle of depression of \(20^\circ\). Find its distance.
4. Find the distance between the two boats.
5. Which boat is closer to the lighthouse?

The end wall of a house is an isosceles triangle. The span is 8 m and the roof rises 2.4 m above the horizontal at the centre.

1. Find the roof angle to the horizontal.
2. Find the rafter length (the sloping side).
3. Tiles cost £14 per square metre. The roof is two rectangles each as long as a rafter and 10 m deep. Find the area and the cost.
4. A different roof has an angle of \(40^\circ\) and the same span. Find its rise.

In a right-angled isosceles triangle with two equal sides of length 1, the hypotenuse is \(\sqrt2\). In an equilateral triangle of side 2 split in half, the height is \(\sqrt3\).

1. Write the exact value of \(\sin45^\circ\) and check it with a calculator.
2. Use the half-equilateral triangle to find the exact values of \(\sin30^\circ\) and \(\cos30^\circ\).
3. Show that \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) for the 5, 12, 13 triangle with \(\theta\) opposite 5.
4. Show that \(\sin^2\theta+\cos^2\theta=1\) for the same triangle.
5. Use the identity to find \(\cos\theta\) if \(\sin\theta=0.6\) and \(\theta\) is acute.