Practice Maths

L15 — Right-Triangle Trigonometry

Key Terms

Right triangle
A triangle containing exactly one 90° angle.
Hypotenuse
The longest side of a right triangle, always opposite the right angle.
Opposite
The side directly across from the reference angle θ.
Adjacent
The side next to the reference angle θ that is not the hypotenuse.
SOH–CAH–TOA
Memory aid: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.
Inverse trig (sin−1, cos−1, tan−1)
Used to find an angle from a known ratio; press shift/inv on a calculator.

SOH – CAH – TOA

For a right triangle with acute angle θ:

θ Adjacent Opposite Hypotenuse A B C
RatioFormulaMemory aid
sin θOpposite ÷ HypotenuseSOH
cos θAdjacent ÷ HypotenuseCAH
tan θOpposite ÷ AdjacentTOA

To find a side: rearrange the ratio. To find an angle: use the inverse (sin²¹, cos²¹, tan²¹).

Exact Values

30°45°60°
sin1/2√2/2√3/2
cos√3/2√2/21/2
tan√3/31√3
Hot Tip: Identify the two sides involved first. If one is the hypotenuse, use sin or cos. If neither is the hypotenuse, use tan. Label opposite and adjacent relative to θ — swapping them is the most common mistake.

Finding a Side

Choose the ratio that links the angle you know to the side you want. Rearrange to isolate the unknown.

Worked Example 1 — Find a side

Find the side opposite to 38° in a right triangle with hypotenuse 15 m.

Step 1: Have θ = 38°, hypotenuse = 15. Want: opposite.

Step 2: sin 38° = opp/hyp ⇒ opp = 15 × sin 38°

Step 3: opp = 15 × 0.6157 ≈ 9.24 m

Finding an Angle

Identify which two sides you know, form the ratio, then apply the inverse trig function.

Worked Example 2 — Find an angle

A right triangle has opposite side 7 and hypotenuse 11. Find the acute angle.

sin θ = 7/11 = 0.6364 ⇒ θ = sin²¹(0.6364) ≈ 39.5°

Exact Values

For 30°–60°–90° and 45°–45°–90° triangles the sides are in fixed integer or surd ratios. Memorise the table in Key Ideas — these appear in non-calculator questions.

Worked Example 3 — Exact value calculation

A right triangle has a 60° angle and hypotenuse 10. Find the exact length of the side opposite 60°.

opp = 10 × sin 60° = 10 × √3/2 = 5√3

Solving a Right Triangle

To “solve” a right triangle means to find all unknown sides and angles. You always need at least one side and one acute angle (or two sides).

  1. Find the unknown side. Fluency

    Give answers to 2 decimal places where not exact.

    • (a) θ = 30°, hypotenuse = 10. Find the opposite side.
    • (b) θ = 45°, hypotenuse = 8. Find the adjacent side.
    • (c) θ = 60°, adjacent = 6. Find the opposite side.
    • (d) θ = 25°, opposite = 5. Find the hypotenuse.

  2. Find the angle. Fluency

    Give answers to 1 decimal place.

    • (a) Opposite = 3, hypotenuse = 6.
    • (b) Adjacent = 5, hypotenuse = 10.
    • (c) Opposite = 4, adjacent = 4.
    • (d) Opposite = 7, hypotenuse = 10.

  3. Exact values. Fluency

    Give exact answers (surds or fractions, not decimals).

    • (a) Evaluate sin 30° + cos 60°.
    • (b) Evaluate tan 45° × cos 45°.
    • (c) Evaluate sin² 60° + cos² 60°.
    • (d) A right triangle has angle 30° and hypotenuse 6. Find the exact length of the side opposite 30°.

  4. Solve the right triangle. Fluency

    Find all unknown sides (to 2 d.p.) and the remaining acute angle.

    • (a) Angle A = 50°, hypotenuse = 12 m.
    • (b) Angle A = 35°, adjacent = 8 cm.

  5. Read from the diagram. Understanding

    The diagram shows a right triangle with angle 40° and hypotenuse 8 m. Angle B is the right angle.

    40° ? ? 8 m A B C
    • (a) Which side (AB or BC) is opposite to the 40° angle?
    • (b) Write the trig equation: sin 40° = ___/8.
    • (c) Find the length of BC (opposite side) to 2 decimal places.
    • (d) Find the length of AB (adjacent side) to 2 decimal places.

  6. Pythagoras and trigonometry combined. Understanding

    A right triangle has legs of length 5 cm and 12 cm.

    • (a) Find the hypotenuse.
    • (b) Find both acute angles, to 1 decimal place.
    • (c) Verify your answer to (b) by confirming the two angles sum to 90°.
    • (d) Use your hypotenuse and the smaller angle to confirm the adjacent side = 12 cm using cos.

  7. Choose the correct ratio. Understanding

    For each situation, state which trig ratio connects the angle to the given sides, then find the unknown.

    • (a) You know θ, the opposite side, and the hypotenuse. Which ratio?
    • (b) You know θ, the adjacent side, and the hypotenuse. Which ratio?
    • (c) You know θ, the opposite side, and the adjacent side. Which ratio?
    • (d) Angle = 55°, adjacent = 9 m. Find the hypotenuse using the correct ratio.

  8. Perimeter of a right triangle. Understanding

    A right triangle has one angle of 38° and a hypotenuse of 20 cm.

    • (a) Find the side opposite to 38°.
    • (b) Find the side adjacent to 38°.
    • (c) Find the perimeter of the triangle.
    • (d) Find the area of the triangle.

  9. Ladder problem. Problem Solving

    A 5 m ladder leans against a vertical wall. The ladder makes an angle of 70° with the ground.

    • (a) How high up the wall does the ladder reach?
    • (b) How far is the base of the ladder from the wall?
    • (c) Safety guidelines say the angle at the base should be at most 75°. At 75°, how high does the same ladder reach?
    • (d) At what angle to the ground must the ladder lean so that it touches the wall at exactly 4 m high?

  10. Sun and shadow. Problem Solving

    A vertical post casts a shadow of 3.2 m when the sun’s angle of elevation is 35°.

    • (a) Draw a diagram showing the post, its shadow, and the angle of elevation of the sun.
    • (b) Write a trigonometric equation relating the post height h to the shadow length and angle.
    • (c) Find the height of the post to 2 decimal places.
    • (d) At what sun angle would the shadow length equal the post height?
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