Exercises: Solving right-angled triangles

In a right-angled triangle $ABC$, the right angle is at $B$, $AB = 6 \text{ cm}$ and $BC = 8 \text{ cm}$. Calculate the length of $AC$, and then write down $\sin A$, $\cos A$ and $\tan A$ as fractions.

Use a calculator to find the following. Give answers correct to two decimal places for parts a) and b), and correct to one decimal place for parts c) and d).

1. a) $\sin 35^\circ$
2. b) $\tan 62^\circ$
3. c) the angle $A$ such that $\cos A = 0.4$
4. d) the angle $B$ such that $\tan B = 1.5$

In a right-angled triangle, the hypotenuse is $12 \text{ cm}$ and one of the acute angles is $40^\circ$.

1. a) Calculate the length of the side opposite the $40^\circ$ angle, correct to one decimal place.
2. b) Calculate the length of the side adjacent to the $40^\circ$ angle, correct to one decimal place.

A ladder $5.5 \text{ m}$ long leans against a vertical wall of a house in Galway. The foot of the ladder is $1.8 \text{ m}$ from the wall on level ground.

1. a) Calculate the angle the ladder makes with the ground, correct to one decimal place.
2. b) Calculate how high up the wall the ladder reaches, correct to two decimal places.

From a point on level ground $45 \text{ m}$ from the base of the Spire in Dublin, the angle of elevation of the top of the Spire is $67.6^\circ$.

1. a) Calculate the height of the Spire, correct to the nearest metre.
2. b) Calculate the straight-line distance from the point on the ground to the top of the Spire, correct to the nearest metre.

A lifeguard stands on a cliff $32 \text{ m}$ above sea level near the Cliffs of Moher. She sees a swimmer at an angle of depression of $18^\circ$.

1. a) Draw a sketch in your copy and mark the angle of depression and the angle of elevation from the swimmer.
2. b) Calculate the horizontal distance from the swimmer to the base of the cliff, correct to the nearest metre.

Two points $A$ and $B$ lie on level ground on opposite sides of a tower $PQ$, where $Q$ is the base of the tower and $A$, $Q$, $B$ lie in a straight line. The angle of elevation of the top $P$ from $A$ is $28^\circ$ and from $B$ is $41^\circ$. The distance $AB$ is $120 \text{ m}$.

1. a) Let $h$ be the height of the tower. Write $AQ$ and $BQ$ in terms of $h$.
2. b) Hence calculate $h$, correct to one decimal place.

A rectangle $PQRS$ has $PQ = 14 \text{ cm}$ and $QR = 9 \text{ cm}$. The diagonal $PR$ is drawn.

1. a) Calculate the angle $RPQ$, correct to one decimal place.
2. b) Calculate the length of the diagonal $PR$ in two different ways, and verify that the answers agree.
3. c) A point $T$ lies on $PQ$ such that angle $TRQ = 30^\circ$. Calculate the length $TQ$, correct to two decimal places.

A pole stands on top of a building. From a point $50 \text{ m}$ from the base of the building, on level ground, the angle of elevation of the bottom of the pole is $35^\circ$ and the angle of elevation of the top of the pole is $39^\circ$.

1. a) Calculate the height of the pole, correct to one decimal place.
2. b) Explain why it would be incorrect to calculate the height of the pole as $50 \tan 4^\circ$.