Practice Maths

Finding Unknown Sides Using Trigonometry

Key Ideas

Key Terms

Step 1
Label the sides (H, O, A) relative to the given angle.
Step 2
Choose the ratio that links the unknown side to a known side.
Step 3
Write the equation, then rearrange to isolate x.
Step 4
Use your calculator — make sure it is in DEGREE mode.
Hot Tip If the unknown x is in the denominator, rearrange before reaching for your calculator. For example, cos(40°) = 8/x rearranges to x = 8 ÷ cos(40°). Getting this step right saves calculation errors every time.

Worked Example

Question: In a right triangle, the angle is 40° and the hypotenuse is 12 cm. Find the opposite side x. Round to 2 decimal places.

Step 1 — Label the sides.
The given angle is 40°. The hypotenuse = 12. The unknown x is the side opposite 40°.

Step 2 — Choose the ratio.
We have opposite (x) and hypotenuse (12) → use sine.

Step 3 — Write and rearrange.
sin(40°) = x / 12
x = 12 × sin(40°)

Step 4 — Calculate.
x = 12 × 0.6428… = 7.71 cm

The Strategy for Finding an Unknown Side

When you know one angle (other than the right angle) and one side length, you can always find any other side using trigonometry. The key is choosing the correct ratio — one that links your known side and unknown side together.

Follow this four-step method every time:

  1. Label the sides relative to your known angle: identify H (hypotenuse), O (opposite), and A (adjacent).
  2. Identify which two sides are involved — the one you know and the one you want.
  3. Choose the correct ratio: SOH if O and H are involved, CAH if A and H, TOA if O and A.
  4. Set up and solve the equation by substituting known values.

Solving When the Unknown is on Top

If your unknown side is in the numerator of the ratio, multiply both sides by the known side. For example, to find the opposite side when you know the hypotenuse and the angle:

sin(θ) = O / H  ⇒  O = H × sin(θ)

Substitute the numbers and use your calculator. Example: θ = 35°, H = 12 cm ⇒ O = 12 × sin(35°) ≈ 6.88 cm.

Solving When the Unknown is on the Bottom

If your unknown side is in the denominator, rearrange first by multiplying both sides by the unknown, then dividing. For example, to find H when you know the opposite and the angle:

sin(θ) = O / H  ⇒  H = O / sin(θ)

Example: θ = 40°, O = 8 m ⇒ H = 8 / sin(40°) ≈ 12.45 m.

Calculator: Degree Mode is Essential

Your calculator must be set to degree mode (not radians) whenever you work with angles in degrees. To check: enter sin(90) — if your calculator gives 1, you are in degree mode. If it gives something like −0.506, switch to degree mode via the settings or MODE menu.

Always round your final answer to an appropriate level of precision — usually 2 decimal places, or as specified in the question.

Exam tip: Always write out the full equation (e.g., sin(35°) = x / 12) before rearranging and calculating. This earns method marks even if your final number is slightly off, and it helps you catch errors in your rearrangement. Never skip straight to a decimal without showing your working.

Mastery Practice

  1. Find the value of x in each right triangle. Round to 2 decimal places. Fluency

    1. Angle = 30°, hypotenuse = 10, find the opposite side x.
      Hint: sin(30°) = x/10
    2. Angle = 45°, hypotenuse = 8, find the adjacent side x.
      Hint: cos(45°) = x/8
    3. Angle = 60°, adjacent = 5, find the opposite side x.
      Hint: tan(60°) = x/5
    4. Angle = 25°, hypotenuse = 14, find the opposite side x.
    5. Angle = 52°, hypotenuse = 9, find the adjacent side x.
    6. Angle = 38°, adjacent = 11, find the opposite side x.
    7. Angle = 70°, opposite = 20, find the hypotenuse x.
      Hint: sin(70°) = 20/x, so x = 20 ÷ sin(70°)
    8. Angle = 15°, adjacent = 7, find the hypotenuse x.
      Hint: cos(15°) = 7/x, so x = 7 ÷ cos(15°)
    9. Angle = 48°, opposite = 6, find the adjacent side x.
    10. Angle = 33°, hypotenuse = 22, find the opposite side x.
    11. Angle = 57°, adjacent = 13, find the hypotenuse x.
    12. Angle = 42°, opposite = 9.5, find the hypotenuse x.

  2. Two-step problems — first label the triangle, then solve. Show all working. Understanding

    1. A right triangle has an acute angle of 34°. The side adjacent to this angle is 16 cm.
      1. Label the three sides relative to 34°.
      2. Which ratio connects the adjacent (16 cm) to the hypotenuse?
      3. Find the hypotenuse. Round to 2 decimal places.
    2. A right triangle has an acute angle of 61°. The hypotenuse is 30 cm.
      1. Find the opposite side.
      2. Find the adjacent side.
      3. Verify your answers using Pythagoras’ theorem: a² + b² should equal 30² = 900.
    3. Two students solve the same problem: angle = 50°, opposite = 8, find hypotenuse x.
      • Sam writes: sin(50°) = 8/x → x = 8/sin(50°) → x ≈ 10.44
      • Rina writes: sin(50°) = x/8 → x = 8 × sin(50°) → x ≈ 6.13
      Who is correct? Explain the error made by the other student.
    4. A right triangle has angle θ = 27° and hypotenuse = 18 cm. Find both remaining sides correct to 2 decimal places, then state which ratio you used for each.

  3. Real-world applications. Problem Solving

    1. A tree casts a shadow on flat ground. The angle of elevation of the sun is 42° and the shadow is 15 m long. The shadow is the adjacent side and the tree height is the opposite side.
      1. Draw and label a right triangle for this situation.
      2. Write the trigonometric equation.
      3. Find the height of the tree. Round to 2 decimal places.
    2. A wheelchair ramp rises to a doorstep 0.9 m above the ground. The ramp makes an angle of 8° with the horizontal.
      1. Which side of the triangle is 0.9 m relative to the 8° angle?
      2. Find the length of the ramp (hypotenuse). Round to 2 decimal places.
      3. Building codes require the ramp length to be at most 7.5 m. Does this ramp comply?
    3. A cable is attached from the top of a 12 m vertical pole to a point on the ground. The cable makes an angle of 55° with the ground.
      1. Find the length of the cable.
      2. Find the distance from the base of the pole to where the cable meets the ground.
    4. A surveyor needs to find the width of a river. She stands at point A directly opposite a tree at point B on the other bank. She walks 40 m along the bank to point C. She measures the angle ACB = 62° (the angle at C in the right triangle formed by A, B, C).
      1. Identify which side is the width of the river (AB) relative to angle C.
      2. Find the width of the river. Round to 2 decimal places.

  4. A ladder leans against a vertical wall. The base of the ladder is 2.5 m from the wall on flat ground. The ladder makes an angle of 72° with the ground. Problem Solving

    Draw it first. Identify the right angle, the 72° angle, and label which side is adjacent (2.5 m), which is opposite (height up the wall) and which is the hypotenuse (ladder length).
    1. Find the length of the ladder. Round to 2 decimal places.
    2. Find how high up the wall the ladder reaches. Round to 2 decimal places.
    3. Safety guidelines say the ladder should reach at least 7 m up the wall. Does this ladder comply?

  5. A roof forms a right triangle in cross-section. The horizontal span (base) of the roof is 10 m and the roof rises at an angle of 24° from the horizontal at each end. Problem Solving

    1. The horizontal run from the wall to the peak is half the total span (5 m). Find the height of the roof peak above the ceiling. Round to 2 decimal places.
    2. Find the length of one sloping side of the roof (the rafter length). Round to 2 decimal places.

  6. From a point on the ground 18 m from the base of a lighthouse, the angle of elevation to the top of the lighthouse is 63°. Problem Solving

    1. Identify which sides of the right triangle are known and unknown.
    2. Find the height of the lighthouse. Round to 2 decimal places.
    3. A boat is at sea level directly opposite the base of the lighthouse. The boat is 120 m from the base of the lighthouse. Find the angle of depression from the top of the lighthouse to the boat. Round to 2 decimal places.

  7. A ship sails on a bearing of North 35° East (i.e. 35° from north, toward east) for 80 km. Problem Solving

    Compass bearings. The 35° angle is measured from north (the vertical). The northward displacement is adjacent to the 35° angle and the eastward displacement is opposite it. The 80 km journey is the hypotenuse.
    1. Find how far north the ship has travelled. Round to 2 decimal places.
    2. Find how far east the ship has travelled. Round to 2 decimal places.

  8. A vertical flagpole casts a shadow 9 m long on flat ground. The angle of elevation of the sun is 48°. Problem Solving

    1. Find the height of the flagpole. Round to 2 decimal places.
    2. Later in the afternoon the shadow lengthens to 14 m. The angle of elevation of the sun has changed. Without recalculating a new angle, use your answer from part (a) to write the new trigonometric ratio and find the new sun angle to 2 decimal places. (Hint: tan(θ) = height ÷ 14)

  9. A surveyor stands at point P on flat ground. She measures two angles of elevation from the same point to the top (T) of a tower. Due to an obstacle, she must approach from two different paths. Problem Solving

    Setup. Path 1: the surveyor stands 25 m from the base, angle of elevation = 55°. Use this to find the tower height. Path 2: the surveyor moves to a new position 40 m from the base (different direction). Use the tower height you found to calculate the angle of elevation from this new position.
    1. Find the height of the tower from Path 1 data. Round to 2 decimal places.
    2. Using that height, find the opposite side if the adjacent distance is 40 m. What trig ratio connects these, and what is its value? Round to 4 decimal places.
    3. State the angle of elevation from the new position. Round to 2 decimal places.

  10. A skatepark designer wants to build a triangular ramp. The ramp surface (hypotenuse) must be exactly 6 m long. The ramp angle at the base must be between 20° and 30° for safety reasons. Problem Solving

    1. Calculate the height of the ramp at the minimum allowed angle of 20°. Round to 2 decimal places.
    2. Calculate the height of the ramp at the maximum allowed angle of 30°. Round to 2 decimal places.
    3. The designer wants the ramp height to be exactly 2.4 m. Calculate the base angle. Round to 2 decimal places. Does this fall within the allowed range?
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