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Grade 10 · Trigonometry · 15 min read

Trigonometry: SOH-CAH-TOA and Every Question Type

Every right-angled trig question in Grade 10 — naming sides, finding a side with sin/cos/tan, finding the hypotenuse, finding an angle with the inverse, special angles, and an angle-of-elevation application — each worked in full, with a diagram and worksheet.

Trigonometry connects the angles of a right-angled triangle to the lengths of its sides. It starts with three ratios — sine, cosine and tangent — and once you can name the sides correctly, every question type follows the same routine.

1Name the three sides

  • The hypotenuse is the longest side, always opposite the right angle.
  • The opposite side is across from the angle you're working with.
  • The adjacent side is next to that angle (but is not the hypotenuse).

Important: 'opposite' and 'adjacent' depend on which angle you're using. Name them fresh for each question.

θadjacentoppositehypotenuse
The sides are named relative to the angle θ. The hypotenuse is always opposite the right angle.

2The memory trick: SOH-CAH-TOA

sinθ = OppositeHypotenuse   cosθ = AdjacentHypotenuse   tanθ = OppositeAdjacent

The routine every time: (1) label the sides, (2) pick the ratio that uses the two sides in play, (3) solve.

3Type 1: Find a side using sine (opposite & hypotenuse)

Worked ExampleExample 1: find x, opposite a 30° angle, hypotenuse 10
  1. 1
    You have the hypotenuse and want the opposite → use sine (SOH).
    Sine links opposite and hypotenuse.
  2. 2
    sin 30° = x / 10.
    Opposite over hypotenuse.
  3. 3
    x = 10 × sin 30° = 10 × 0,5 = 5.
    Multiply both sides by 10.

4Type 2: Find a side using cosine or tangent

Worked ExampleExample 2: find x, adjacent to a 40° angle, hypotenuse 12
  1. 1
    Adjacent and hypotenuse → use cosine (CAH).
    cos = adj/hyp.
  2. 2
    cos 40° = x / 12 → x = 12 × cos 40°.
    Isolate x.
  3. 3
    x ≈ 12 × 0,766 = 9,19.
    Use the calculator (in degrees).
Worked ExampleExample 3: find x, opposite a 35° angle, adjacent = 8 (use tan)
  1. 1
    Opposite and adjacent → use tangent (TOA).
    tan = opp/adj.
  2. 2
    tan 35° = x / 8 → x = 8 × tan 35°.
  3. 3
    x ≈ 8 × 0,700 = 5,60.

5Type 3: Find the hypotenuse

Worked ExampleExample 4: the side opposite a 25° angle is 6; find the hypotenuse
  1. 1
    Opposite and hypotenuse → sine: sin 25° = 6 / h.
    The unknown is on the bottom this time.
  2. 2
    Rearrange: h = 6 / sin 25°.
    Multiply both sides by h, divide by sin 25°.
  3. 3
    h ≈ 6 / 0,4226 = 14,2.

6Type 4: Find an angle (use the inverse)

When you know two sides and want the angle, use the inverse function (sin−1, cos−1, tan−1 on your calculator).

Worked ExampleExample 5: opposite = 7, hypotenuse = 10; find θ
  1. 1
    Opposite and hypotenuse → sine: sin θ = 710 = 0,7.
    Set up the ratio first.
  2. 2
    θ = sin−1(0,7).
    Inverse sine 'undoes' the sine to give the angle.
  3. 3
    θ ≈ 44,4°.
    Use the sin−1 (or 2ndF sin) key.

7Type 5: Special angles (exact values)

Memorise these — they come up without a calculator: sin 30° = 12, cos 60° = 12, sin 60° = cos 30° = 32, sin 45° = cos 45° = 12, tan 45° = 1.

8Type 6: Application: angle of elevation

θdistanceheightline of sight
The angle of elevation θ is measured upward from the horizontal to the line of sight.
Worked ExampleExample 6: a ladder reaches 4 m up a wall at 60° to the ground; how long is it?
  1. 1
    The 4 m is opposite the 60° angle; the ladder is the hypotenuse → sine.
    opp and hyp → SOH.
  2. 2
    sin 60° = 4 / L → L = 4 / sin 60°.
    The unknown hypotenuse is on the bottom.
  3. 3
    L ≈ 4 / 0,866 = 4,62 m.
💡

Finding a side? Use sin/cos/tan directly. Finding an angle? Use the inverse (sin−1 etc). And always keep your calculator in DEGREES — wrong mode is the #1 cause of 'random' wrong answers.

⚠️

Pick the ratio from the two sides you have. Don't default to sine — if you have opposite and adjacent, it's tangent. Label the sides first, every time.

Trig is a big, reliable section of Paper 2 in Grades 11–12. Master these six question types now and the harder work later builds on solid ground.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet✅ Memo📝 Exam paper🗝️ Marking guide
  1. 1
    Find x opposite a 30° angle, hypotenuse 8. (sine)
    x830°
    Show answer ▾
    x = 8 sin30° = 8×0.5 = 4
  2. 2
    Find x opposite a 50° angle, hypotenuse 10. (sine)
    x1050°
    Show answer ▾
    10 sin50° ≈ 7.66
  3. 3
    Find x adjacent to a 60° angle, hypotenuse 14. (cosine)
    x1460°
    Show answer ▾
    14 cos60° = 14×0.5 = 7
  4. 4
    Find x opposite a 40° angle, adjacent 9. (tangent)
    9x40°
    Show answer ▾
    9 tan40° ≈ 7.55
  5. 5
    The side opposite a 30° angle is 5. Find the hypotenuse. (hypotenuse)
    5?30°
    Show answer ▾
    h = 5/sin30° = 10
  6. 6
    opposite = 6, hypotenuse = 12. Find θ. (angle)
    612θ
    Show answer ▾
    sinθ = 0.5 → θ = 30°
  7. 7
    adjacent = 8, hypotenuse = 16. Find θ. (angle)
    816θ
    Show answer ▾
    cosθ = 0.5 → θ = 60°
  8. 8
    opposite = 5, adjacent = 5. Find θ. (angle)
    55θ
    Show answer ▾
    tanθ = 1 → θ = 45°
  9. 9
    Give the exact value of sin 30° and tan 45°. (special)
    Show answer ▾
    12 and 1
  10. 10
    A rope to the top of a 6 m pole makes 40° with the ground. Rope length? (application)
    Show answer ▾
    sin40° = 6L → L = 6/sin40° ≈ 9,33 m

Now practise it

Download Grade 10 past papers and worksheets on this topic.

Go to Grade 10 papers →

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