Practice Maths

Finding Unknown Angles Using Trigonometry

Key Ideas

Key Terms

inverse trig function
.
Check
Both acute angles in a right triangle must add up to 90°.
Hot Tip “Inverse” means “undo.” If sin(30°) = 0.5, then sin⊃⁻¹(0.5) = 30°. The inverse undoes the trig function to give you the angle back. Always calculate the fraction first, then apply the inverse trig key.

Worked Example

Question: In a right triangle, the opposite side = 5 and the hypotenuse = 10. Find θ.

Step 1 — Identify the ratio.
We have opposite (5) and hypotenuse (10) → use sine.

Step 2 — Write the equation.
sin θ = 5/10 = 0.5

Step 3 — Apply the inverse.
θ = sin⊃⁻¹(0.5) = 30°

Step 4 — Check.
The other acute angle = 90° − 30° = 60°. ✓

From Ratios to Angles: Inverse Trig Functions

When you know two sides of a right-angled triangle and need to find an angle, you work backwards through the trig ratio. The inverse trig functions do exactly this — they take a ratio and return the angle.

  • sin−1(x) gives the angle whose sine is x
  • cos−1(x) gives the angle whose cosine is x
  • tan−1(x) gives the angle whose tangent is x

These are sometimes written as arcsin, arccos, and arctan, and on your calculator they appear as 2nd + sin, cos, or tan (the shifted function).

The Step-by-Step Method

  1. Label the three sides relative to the unknown angle θ: identify H, O, and A.
  2. Write a trig equation using the two sides you know. For example, if you know O and H: sin(θ) = O / H.
  3. Substitute the numbers. For example: sin(θ) = 5 / 13.
  4. Apply the inverse function: θ = sin−1(5/13) ≈ 22.6°.

You can enter 5 ÷ 13 on your calculator first, or type sin−1(5/13) directly — both give the same result as long as your calculator is in degree mode.

Choosing Which Inverse Function to Use

The choice depends on which two sides you know relative to the angle you want to find:

  • Know O and H ⇒ use sin−1(O/H)
  • Know A and H ⇒ use cos−1(A/H)
  • Know O and A ⇒ use tan−1(O/A)

This is the same SOH CAH TOA choice as before — you just apply the inverse at the end instead of multiplying or dividing.

Rounding and Checking Your Answer

Angles in right-angled triangles must be between 0° and 90° (exclusive). If your calculator gives you a negative angle or one larger than 90°, recheck your ratio — you likely have O and A swapped, or the calculator is in radian mode.

Round to one or two decimal places unless otherwise specified. You can verify: the two non-right angles must add to exactly 90° (since all three angles sum to 180°).

Key tip: The expression sin−1 does NOT mean 1/sin(θ) — it means "the angle whose sine is...". Students often confuse this notation. On your calculator, press 2nd (or SHIFT) then sin to access sin−1. It is a completely different operation from reciprocating the sine.

Mastery Practice

  1. Find angle θ in each right triangle. Round to 1 decimal place. Fluency

    1. Opposite = 4, hypotenuse = 8. Find θ.
    2. Adjacent = 6, hypotenuse = 10. Find θ.
    3. Opposite = 7, adjacent = 7. Find θ.
    4. Opposite = 3, hypotenuse = 5. Find θ.
    5. Adjacent = 9, hypotenuse = 15. Find θ.
    6. Opposite = 12, adjacent = 5. Find θ.
    7. Opposite = 8, hypotenuse = 17. Find θ.
    8. Adjacent = 4.5, hypotenuse = 9. Find θ.
    9. Opposite = 11, adjacent = 14. Find θ.
    10. Opposite = 6.2, hypotenuse = 13.0. Find θ.
    11. Adjacent = 20, hypotenuse = 25. Find θ.
    12. Opposite = 15, adjacent = 8. Find θ.

  2. Identify which inverse to use and verify your answer. Understanding

    1. A right triangle has sides 8 (opposite) and 15 (adjacent). Liam says he should use cos⊃⁻¹(8/15) to find θ.
      1. Is Liam correct? Explain which inverse function to use and why.
      2. Find θ using the correct function.
    2. A right triangle has opposite = 9 and hypotenuse = 41.
      1. Find θ to 1 decimal place.
      2. Use your value of θ to find the adjacent side using a different trig ratio.
      3. Verify using Pythagoras’ theorem.
    3. Explain why there are two different inverse trig functions that could give you the same answer in special cases. Give an example using a 45°-45°-90° triangle.
    4. A right triangle has the three sides: 5, 12, 13. Without a calculator, write the exact values of all three trig ratios for the angle opposite the side of length 5. Then find that angle using the inverse function.

  3. Angle of elevation and angle of depression problems. Problem Solving

    1. From a point 30 m from the base of a building, the top of the building is at an angle of elevation. The building is 45 m tall.
      1. Draw and label the right triangle.
      2. Find the angle of elevation to 1 decimal place.
    2. A lifeguard in a tower 8 m above sea level spots a swimmer. The horizontal distance from the base of the tower to the swimmer is 22 m. Find the angle of depression from the lifeguard’s eyes down to the swimmer. Round to 1 decimal place.
    3. A ski run descends 420 m vertically over a horizontal distance of 1260 m. Find the angle the ski slope makes with the horizontal. Round to 1 decimal place.
    4. From the top of a 60 m cliff, two boats are seen in the sea. Boat A has an angle of depression of 25° and Boat B has an angle of depression of 40°. Both boats are in a straight line from the base of the cliff.
      1. Find the distance from the base of the cliff to Boat A.
      2. Find the distance from the base of the cliff to Boat B.
      3. How far apart are the two boats?

  4. A car drives up a hill. The road rises 8 m vertically over a horizontal distance of 60 m. Problem Solving

    1. Draw a right triangle representing this situation. Label the rise (opposite), run (adjacent) and road length (hypotenuse).
    2. Find the angle the road makes with the horizontal. Round to 1 decimal place.
    3. The road continues and rises a further 5 m over 35 m horizontally. Find the angle of this second section. Is it steeper or gentler than the first section?

  5. A builder needs to cut roof rafters. The roof peak is 3.2 m above the ceiling and the horizontal run from the wall to the peak is 6 m. Problem Solving

    1. Find the pitch angle (angle between the rafter and the horizontal ceiling). Round to 1 decimal place.
    2. The builder must cut the bottom end of each rafter at exactly this pitch angle. If the builder accidentally uses the complementary angle (90° − your answer), how much error is introduced in degrees?

  6. From two points on the ground both directly in line with the base of a tower, the angles of elevation to the top are measured. Point A is 20 m from the base; Point B is 50 m from the base. The tower is h metres tall. Problem Solving

    Two-step problem. You need to first find the tower height from the information given (tower height = 28 m), then calculate each angle of elevation using the inverse tangent.
    1. Given the tower height is 28 m, find the angle of elevation from Point A. Round to 1 decimal place.
    2. Find the angle of elevation from Point B. Round to 1 decimal place.
    3. How many degrees greater is the angle from Point A compared to Point B?

  7. A plane flies 240 km east and then 180 km north, arriving at its destination. Find the bearing (measured clockwise from north) of the destination from the starting point. Problem Solving

    Bearings. The bearing is measured clockwise from north. The angle you find with inverse tan is the angle from the east axis. Convert it to a bearing by subtracting from 90°, or work directly from north using the northward and eastward displacements.
    1. Draw a right triangle with the eastward leg (240 km) and northward leg (180 km). Label the angle at the origin.
    2. Find the angle the flight path makes with north. Round to 1 decimal place.
    3. State the bearing of the destination from the starting point.

  8. A triangular plot of land has a right angle at corner C. Side CA = 45 m and side CB = 60 m. Find all unknown angles of the triangle. Problem Solving

    1. Use the two given sides to find angle A (at corner A). Round to 1 decimal place.
    2. Find angle B using the angle sum of a triangle. Verify by also calculating it directly with an inverse trig function.
    3. Calculate the hypotenuse AB. Round to 2 decimal places.

  9. A coastguard on a cliff 35 m above sea level spots two objects. A buoy directly below at sea level and a yacht further out to sea. The horizontal distance to the yacht is 120 m and the yacht’s deck is at sea level. Problem Solving

    1. Find the angle of depression from the coastguard to the yacht. Round to 1 decimal place.
    2. A rescue boat travels from the buoy directly below the coastguard out to the yacht. What angle does the rescue boat’s path make with the base of the cliff? Round to 1 decimal place.
    3. Check: the angle of depression and the angle the rescue boat path makes with the horizontal should be equal (alternate interior angles). Do your answers confirm this?

  10. An architect is designing a triangular feature wall. The wall has a horizontal base of 8 m and a vertical height of 5 m, creating a right triangle with the right angle at the bottom-left corner. Problem Solving

    1. Find the acute angle at the bottom-right corner (the base angle). Round to 1 decimal place.
    2. The architect wants to add a diagonal support beam from the bottom-right corner to a point on the vertical side, 2 m above the floor. Find the angle this beam makes with the horizontal. Round to 1 decimal place.
    3. What is the length of that support beam? Round to 2 decimal places. (Hint: use trigonometry with the angle you found, or Pythagoras.)
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