Mathematics • Year 9 • Unit 3 • Lesson 10

Mixed — Finding Sides with Any Ratio

Make the trig choice automatic: list the two sides → pick the matching ratio (opp+hyp → sin, adj+hyp → cos, opp+adj → tan) → rearrange → compute. Build from a fully-worked example through guided fill-ins to eight independent mixed problems.

Build · I Do / We Do / You Do

1. I do — fully worked example

A right triangle has θ = 50° and hyp = 14. Find opp. Watch every step — especially the "list the sides" step, which is the heart of L10.

Problem. hyp = 14, θ = 50°, opp = ? Round to 2 d.p. DEG mode (sin 30° must give 0.5).

Step 1 — List the two sides involved.

Known: hyp = 14. Wanted: opp. So the two sides involved are opp and hyp.

Reason: this is the FIRST step in any trig side problem. Pin down the pair before doing anything else.

Step 2 — Match the pair to the ratio.

opp + hyp → sin (SOH).

Reason: SOH = Sine = Opp/Hyp. This is the only ratio with that exact pair.

Step 3 — Set up the equation.

sin 50° = opp / 14

Reason: substitute the known hyp and the wanted opp into sin θ = opp/hyp.

Step 4 — Rearrange to isolate opp.

opp = 14 × sin 50°

Reason: opp is on top → multiply both sides by the bottom (14).

Step 5 — Compute and check.

opp ≈ 14 × 0.7660 ≈ 10.72

Sense check: opp should be SMALLER than hyp (always). 10.72 < 14 ✓.

Answer: opp ≈ 10.72.

Stuck? Revisit lesson § "Watch Me Solve It · Choose sin or cos" — same setup, same numbers.

2. We do — fill in the missing steps

A right triangle has θ = 28°, opp = 9. Find adj. Fill in each blank. 4 marks

Step 1 — List the two sides involved. Known: opp = ____ . Wanted: ________ . Two sides involved: opp and adj .

Step 2 — Match the pair to the ratio. opp + adj → __________ (acronym: ________ ).

Step 3 — Set up:

tan ____ ° = ____ / adj

Step 4 — Rearrange to put adj on its own:

adj = ____ / tan ____ °

Step 5 — Compute:

adj ≈ 9 / ________ ≈ ________

Stuck? Revisit lesson § "Watch Me Solve It · Tan is needed" — same setup, same numbers.

3. You do — independent practice

For each problem, write the side-pair, name the ratio, then compute. Round to 2 d.p. unless told.

Foundation — one ratio, one rearrangement

3.1 θ = 40°, hyp = 18. Find adj. (Side-pair: adj + hyp → ?) 1 mark

3.2 θ = 30°, opp = 5. Find hyp. (Side-pair: opp + hyp → ?) 1 mark

3.3 θ = 55°, adj = 7. Find opp. (Side-pair: opp + adj → ?) 1 mark

3.4 θ = 60°, hyp = 22. Find opp. 1 mark

Standard — name the ratio AND solve

3.5 θ = 45°, opp = 8. Find adj AND hyp. (Hint: for a 45° angle, opp = adj.) 2 marks

3.6 θ = 38°, adj = 11. Find hyp. 2 marks

Extension — push your thinking

3.7 θ = 32°, hyp = 10. Find BOTH legs (opp and adj). Then verify Pythagoras: opp² + adj² should ≈ 100. 3 marks

3.8 A right triangle has one acute angle θ. You are told opp = 6 and want to find adj — but you ONLY know that the triangle is a 3-4-5-type triangle scaled by some factor (so the angles match a 3-4-5). Using just trig ratios (no Pythagoras), find adj. 2 marks

Stuck on 3.8? In any 3-4-5 triangle with θ opposite the 3-side, tan θ = 3/4. So adj = opp / tan θ = 6 / (3/4) = 8.

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What I'll revisit before next class:

Answers — Do not peek before attempting

Section 2 — We do (θ = 28°, opp = 9, find adj)

Step 1: Known opp = 9. Wanted adj. Two sides: opp and adj.
Step 2: opp + adj → tan (acronym TOA).
Step 3: tan 28° = 9 / adj.
Step 4: adj = 9 / tan 28°.
Step 5: adj ≈ 9 / 0.5317 ≈ 16.93.

3.1 — θ = 40°, hyp = 18, find adj

adj + hyp → cos. adj = 18 cos 40° ≈ 18 × 0.7660 ≈ 13.79.

3.2 — θ = 30°, opp = 5, find hyp

opp + hyp → sin. hyp = 5 / sin 30° = 5 / 0.5 = 10 exactly.

3.3 — θ = 55°, adj = 7, find opp

opp + adj → tan. opp = 7 tan 55° ≈ 7 × 1.4281 ≈ 10.00.

3.4 — θ = 60°, hyp = 22, find opp

opp + hyp → sin. opp = 22 sin 60° ≈ 22 × 0.8660 ≈ 19.05.

3.5 — θ = 45°, opp = 8, find adj and hyp

tan 45° = 1, so adj = opp / 1 = 8 exactly.
hyp = opp / sin 45° = 8 / 0.7071 ≈ 11.31.
For 45°: opp = adj (isosceles right triangle); hyp = leg × √2.

3.6 — θ = 38°, adj = 11, find hyp

adj + hyp → cos. hyp = 11 / cos 38° ≈ 11 / 0.7880 ≈ 13.96.

3.7 — θ = 32°, hyp = 10, both legs

opp = 10 sin 32° ≈ 10 × 0.5299 ≈ 5.30.
adj = 10 cos 32° ≈ 10 × 0.8480 ≈ 8.48.
Pythagoras: 5.30² + 8.48² ≈ 28.09 + 71.91 ≈ 100.00 = 10² ✓.

3.8 — 3-4-5 triangle, opp = 6, find adj (trig only)

In any 3-4-5 triangle with θ opposite the 3-side, tan θ = 3/4 = 0.75.
adj = opp / tan θ = 6 / 0.75 = 8 exactly.
This makes sense: a 3-4-5 triangle scaled by 2 is the 6-8-10 triangle, so opp = 6 pairs with adj = 8.