Practice Maths

Applications of Trigonometry

Key Ideas

Key Terms

Angle of elevation
Measured upward from the horizontal to the line of sight.
Angle of depression
Measured downward from the horizontal to the line of sight.
alternate angles
They are equal when the horizontal lines are parallel.
Bearings
Measured clockwise from North (000° to 360°).
Hot Tip The angle of elevation (measured from the ground up) is equal to the angle of depression (measured from the top down) when both horizontal lines are parallel. This is a very common exam shortcut — draw parallel horizontal lines to see the alternate angles clearly.

Worked Example — Lighthouse

Question: A lighthouse is 35 m tall. A boat at sea observes the top of the lighthouse at an angle of elevation of 18°. Find the horizontal distance from the boat to the base of the lighthouse. Round to the nearest metre.

Step 1 — Draw and label.
Right triangle: lighthouse height = opposite = 35 m. Horizontal distance = adjacent = d. Angle at the boat = 18°.

Step 2 — Choose the ratio.
tan(18°) = opposite / adjacent = 35 / d

Step 3 — Solve for d.
d = 35 ÷ tan(18°) = 35 ÷ 0.3249 = 107.7 m ≈ 108 m

Always Draw a Diagram

Application problems describe a real-world situation in words. Your first job is always to draw a clear diagram, label all the given information, and mark the unknown you need to find. This transforms a confusing word problem into a familiar right-angled triangle problem.

Key things to mark on your diagram: the right angle, all known lengths, all known angles, and a question mark for the unknown.

Angles of Elevation and Depression

These are the most common application type in Year 9:

  • Angle of elevation: the angle measured upward from the horizontal to a point above you. Imagine looking up at the top of a building — the angle between the ground and your line of sight is the angle of elevation.
  • Angle of depression: the angle measured downward from the horizontal to a point below you. Imagine standing on a cliff and looking down at a boat — the angle between the horizontal and your line of sight is the angle of depression.

Key fact: the angle of elevation from A to B is always equal to the angle of depression from B to A (alternate interior angles with parallel horizontals).

Common Real-World Setups

Typical problems involve:

  • A ladder leaning against a wall — the ladder is the hypotenuse, the wall is opposite the ground angle, the ground distance is adjacent.
  • A ramp — the ramp surface is the hypotenuse, the height is opposite the base angle, the horizontal run is adjacent.
  • A tower or flagpole — an observer at ground level looks up at the top; the height is the opposite side, the horizontal distance is adjacent, and the angle of elevation is θ.
  • Navigation bearings — converting compass directions to right-triangle components. A bearing of N30°E means 30° east of north, forming a right triangle with north/east as the legs.

Multi-Step Problems

Some problems require two (or more) separate trig calculations. Work systematically: solve the first triangle completely, then use that answer as input for the second triangle. Avoid rounding intermediate answers — carry full calculator precision through until the final step, then round once at the end.

Exam strategy: In application problems, always state the angle and the sides involved before writing the trig equation. Write something like: "Using tan, opposite = height, adjacent = 15 m, angle = 32°", then write tan(32°) = h/15. Examiners reward clear, logical working — a correct method with an arithmetic slip still earns most of the marks.

Mastery Practice

  1. Angle of elevation and depression — find the unknown measurement. Round to 2 decimal places. Fluency

    1. The angle of elevation to the top of a building is 34° from a point 50 m from its base. Find the height of the building.
    2. A kite is flying at a height of 40 m. The string makes an angle of elevation of 52° with the ground. Find the length of the string.
    3. From the top of a 20 m cliff, a boat is seen at an angle of depression of 15°. Find the horizontal distance to the boat.
    4. A person standing 80 m from a tower observes the top at an angle of elevation of 27°. Find the height of the tower.
    5. A plane flying at 1500 m altitude begins its descent. The pilot observes the runway at an angle of depression of 3°. Find the horizontal distance from the plane to the runway.
    6. From a window 12 m above the ground, a car on the road below is seen at an angle of depression of 40°. Find the horizontal distance from the building to the car.

  2. Bearings and direction problems. Understanding

    1. A ship travels 120 km on a bearing of 040° (N40°E). This means it travels at 40° east of north.
      1. Draw a right triangle to represent this journey, with North pointing up.
      2. Find how far north the ship has travelled (the northward component).
      3. Find how far east the ship has travelled (the eastward component).
    2. A helicopter flies 85 km on a bearing of 310° (N50°W). Find the northward and westward components of its journey. Round to 1 decimal place.
    3. From point A, a hiker walks 6.4 km due east to point B, then 4.8 km due north to point C.
      1. Find the straight-line distance AC.
      2. Find the bearing from A to C (the angle measured clockwise from north).
    4. Two ships leave port at the same time. Ship P travels on a bearing of 060° for 50 km. Ship Q travels on a bearing of 150° for 50 km. Find the distance between the two ships.

  3. Multi-step and mixed applications. Problem Solving

    1. A surveyor stands between two buildings A and B on opposite sides of a road. She stands 15 m from the base of Building A and 22 m from the base of Building B. Building A has a roof at an angle of elevation of 58° from her position. Building B has a roof at an angle of depression of 32° from its roof down to her position.
      1. Find the height of Building A.
      2. Find the height of Building B.
      3. What is the difference in heights between the two buildings?
    2. A hiker at the base of a hill observes the summit at an angle of elevation of 22°. She walks 300 m directly up the slope (the hypotenuse) to reach the summit.
      1. Find the vertical height gained.
      2. Find the horizontal distance covered.
      3. She descends directly down the other side, which has a slope angle of 18°. If she walks until she is back at the same altitude as the base, find the length of the descent path (hypotenuse on the other side).
    3. A satellite dish is mounted on a roof 9 m above the ground. A technician stands 12 m from the base of the building looking up at the dish.
      1. Find the straight-line distance from the technician to the dish.
      2. Find the angle of elevation from the technician to the dish.
      3. If the dish needs to point to a satellite at an angle of elevation of 65° from the dish, and the dish is a circular reflector of diameter 1.2 m, what is the minimum clear height the signal path needs above the dish? (Find how high directly above the dish the signal travels over a horizontal distance of 2 m.)
    4. From the roof of a 30 m building, the angles of depression to the near and far edges of an adjacent car park are 55° and 28° respectively.
      1. Find the horizontal distance to the near edge of the car park.
      2. Find the horizontal distance to the far edge.
      3. Hence find the length (depth) of the car park.

  4. Real-world trigonometry applications. Problem Solving

    1. A ladder 5 m long leans against a wall at an angle of elevation of 65°. How high does the ladder reach up the wall? Round to 2 decimal places.
    2. A 10 m ramp rises to a loading dock. The ramp makes an angle of elevation of 8° with the ground. Find the height of the loading dock above the ground. Round to 2 decimal places.
    3. A zip-line cable is anchored at the top of a 14 m pole and is attached to the ground at a point 20 m horizontally from the base. Find the angle of depression from the top of the pole to the ground anchor. Round to 1 decimal place.

  5. Bearings and navigation problems. Problem Solving

    1. A coastguard station C observes a vessel V on a bearing of 125° at a distance of 40 km. How far south and how far east of the coastguard station is the vessel? Round to 1 decimal place.
    2. A yacht sails 30 km on a bearing of 200° (S20°W).
      1. Find the southward component of the journey.
      2. Find the westward component of the journey.
      3. To return directly to the starting point, what bearing must the yacht sail, and how far is the return trip?
    3. Two aircraft leave the same airport simultaneously. Aircraft A flies on a bearing of 070° for 200 km. Aircraft B flies on a bearing of 340° for 150 km. Find the straight-line distance between the two aircraft at that moment. Round to the nearest km.

  6. Angles of elevation with changing position. Problem Solving

    1. A person observes the top of a lighthouse at an angle of elevation of 20°. They walk 50 m closer and the angle of elevation is now 35°. Find the height of the lighthouse. (Let h = height and d = original distance; set up two equations in h and d.) Round to 1 decimal place.
    2. From the top of a vertical cliff 80 m high, a fishing boat is spotted at an angle of depression of 12°. Ten minutes later the same boat is at an angle of depression of 25°. How far has the boat travelled in those ten minutes? Round to the nearest metre.

  7. Three-dimensional trigonometry applications. Problem Solving

    1. A rectangular swimming pool is 25 m long and 10 m wide. A diagonal rope is stretched from one corner at the water surface across to the diagonally opposite corner. Find the length of the rope and the angle it makes with the longer side. Round to 2 decimal places.
    2. A flagpole stands vertically on a hill. From a point A at the base of the hill, the angle of elevation to the base of the flagpole is 15°. The horizontal distance from A to the base of the flagpole is 60 m. The flagpole is 8 m tall.
      1. Find the height of the base of the flagpole above point A.
      2. Find the angle of elevation from A to the top of the flagpole.

  8. Multi-step elevation and depression problems. Problem Solving

    1. A person at point P looks up to the top of Tower A at an angle of elevation of 42°. Tower A is 30 m tall. From the top of Tower A, they look up to the top of Tower B at an angle of elevation of 18°. The horizontal distance from Tower A to Tower B is 50 m.
      1. Find the horizontal distance from P to Tower A.
      2. Find the additional height of Tower B above Tower A.
      3. Find the total height of Tower B.
    2. A surveyor standing at ground level at point S measures the angle of elevation to the top of a hill as 14°. She walks 200 m directly toward the hill along flat ground and now measures the angle of elevation as 23°. Calculate the height of the hill. Round to the nearest metre.

  9. Return-journey and triangular bearing problems. Problem Solving

    1. A rescue helicopter leaves base on a bearing of 055° and travels 80 km to reach an injured hiker at point H. It then flies due west 60 km to a fuel depot at point D.
      1. Find the northward and eastward displacement from base to H.
      2. Find the coordinates of D relative to the base (taking north as positive y, east as positive x).
      3. Find the straight-line distance and bearing from D back to the base.
    2. A triangular sailing race course has legs AB, BC, and CA. A is the start. B is 12 km from A on a bearing of 030°. C is 9 km from B on a bearing of 120°. Find the distance and bearing from C back to A to complete the course. Round distances to 1 decimal place and bearings to the nearest degree.

  10. Shadow and angle-of-elevation investigations. Problem Solving

    1. At a certain time of day the sun’s rays strike the ground at an angle of elevation of 38°. A vertical building casts a shadow 45 m long. Find the height of the building. Round to 2 decimal places.
    2. Two vertical posts PQ and RS are 4 m and 7 m tall respectively and stand 10 m apart on flat ground. A rope is tied from the top of each post to the base of the other. Find where the ropes intersect above the ground — specifically, the height above the ground at the crossing point. (Hint: use similar triangles.)
    3. The angle of elevation of the sun changes from 25° to 60° between 8 am and noon. A 10 m vertical pole is standing in flat ground. Find the difference in the lengths of the shadow cast at 8 am and at noon. Round to 2 decimal places.
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