Trigonometry with acute angles
Use sine, cosine and tangent ratios to find unknown sides and angles in right-angled triangles; work with exact trig values for 30°, 45° and 60°; and solve multi-step problems involving bearings and angles of elevation and depression.
Worked examples
Finding an unknown side using the sine ratio
Straightforward
Problem
In a right-angled triangle, the hypotenuse is cm and one angle is . Find the side opposite the angle, to 2 decimal places.
1
Identify the known and unknown sides relative to the given angle.
Known: hypotenuse cm, angle . Unknown: opposite side .
2
Write the sine ratio connecting the opposite side and the hypotenuse.
3
Solve for by multiplying both sides by .
Answer
cm
Using exact trig values to find a northward distance from a bearing
Moderate
Problem
A ship sails on a bearing of for km from port . Find the exact northward distance the ship travels.
1
Draw a diagram. A bearing of means the ship travels clockwise from north.
The angle between the ship's path and the north direction is .
2
Identify which trig ratio gives the northward component. The northward distance is the side adjacent to the angle.
3
Substitute the exact value and simplify.
Answer
The northward distance is km (approximately km).
Multi-step navigation using Pythagoras and trigonometry
Challenging
Problem
A boat travels km due east from port to point , then km due north to point . Find the distance and the bearing of from , to the nearest degree.
1
Find the direct distance using Pythagoras' theorem. The eastward leg and northward leg are the two shorter sides of a right-angled triangle.
2
To find the bearing of from , identify the angle that makes with north. The eastward leg is opposite and the northward leg is adjacent.
3
State the bearing. Since is north-east of , the bearing is degrees east of north.
Answer
km and the bearing of from is T.
Practise
Q1·Straightforward
In a right-angled triangle, the hypotenuse is cm and one angle is . Find the length of the side opposite the angle, to 2 decimal places.
Explanation
Using the sine ratio: , so cm.
Q2·Straightforward
In a right-angled triangle, the side adjacent to a angle is m. Find the length of the side opposite the angle, to 2 decimal places.
Explanation
Using the tangent ratio: , so m.
Q3·Straightforward
In a right-angled triangle, the side opposite an unknown angle is cm and the hypotenuse is cm. Find the unknown angle to the nearest degree.
Explanation
, so .
Q4·Straightforward
A ramp rises at an angle of to the horizontal. The horizontal distance along the ground is m. Find the length of the ramp surface, to 2 decimal places.
Explanation
, so m.
Q5·Moderate
Without using a calculator, find the exact value of . Give your answer as a fraction.
/
Explanation
Using exact values: and . So .
Q6·Moderate
A ship departs from port on a bearing of and sails km. How far north of the port is the ship, to 2 decimal places?
Explanation
The northward component km.
Q7·Moderate
From a point m from the base of a water tower, the angle of elevation of the top of the tower is . Find the height of the tower, to 2 decimal places.
Explanation
, so m.
Q8·Moderate
Find the exact value of .
Explanation
.
Q9·Challenging
From point , a lighthouse is observed on a bearing of . From point , which is m due east of , the lighthouse is observed on a bearing of . Find the distance , to the nearest metre.
Explanation
Place at the origin and at . From on bearing : . From on bearing (i.e. west of north): . Equating -coordinates: , so . Equating -coordinates and substituting: . Solving: , so m.
Q10·Challenging
From a boat at sea level, the top of a m lighthouse is observed at an angle of elevation of . A marker buoy is anchored on the sea floor directly below the lighthouse at a depth of m. Find the straight-line distance from the boat to the buoy, to 2 decimal places.
Explanation
Horizontal distance: m. The buoy is m below the sea surface directly under the lighthouse, so by Pythagoras: m.
Q11·Challenging
From lighthouse , ship is on a bearing of at a distance of km, and ship is on a bearing of at a distance of km. Find the distance between the two ships, to 2 decimal places.
Explanation
The angle at between bearings and is . Since km and angle , triangle is right-angled and isoceles. By Pythagoras: km.
Q12·Challenging
From a point m from the base of a building, a surveyor measures the angle of elevation to the roof as and to the top of a vertical antenna on the roof as . Find the height of the antenna, to 2 decimal places.
Explanation
Height of roof m. Height of antenna top m. Height of antenna m.