Lesson 10 ~25 min Unit 3 · Trigonometry +85 XP

Mixed — Finding Sides with Any Ratio

Decide between sin, cos and tan based on which two sides are involved. A simple decision flowchart turns any problem into a solved one.

Today's hook: Four right triangles, four different known/unknown combinations of sides. One strategy — list the two sides, pick the matching ratio — works every time.

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Think First

warm-up

A triangle has angle $\theta = 35°$, hyp = 12, and you want adj. Which ratio uses adj and hyp? Pick it, then rearrange.

Record your answer in your workbook.

The Big Idea

+5 XP

Picking sin vs cos vs tan is the single most-tested skill in trigonometry. The trick: ignore everything else — just look at the TWO sides involved.

opp + hyp → sin. adj + hyp → cos. opp + adj → tan. Three combinations, three ratios — that's it. Label, list, decide, rearrange, compute.

Pick the ratio matching the two sides involved

List the pair

Always write down which two sides are involved before choosing a ratio.

Three options only

There are only three pairings — three ratios.

Rearrange then compute

Set up first, isolate the unknown, then press the calculator.

What You'll Master

objectives

Know

The three side-pairings and their ratios
SOH-CAH-TOA used in reverse to choose
Each rearrangement direction

Understand

How to read a diagram to identify which sides are known/wanted
Why the choice of ratio depends ONLY on the sides involved
How to verify with Pythagoras after computing

Can Do

Identify and apply the correct ratio in any single-step problem
Solve for any single side given an angle and another side
Switch between ratios fluently

Words You Need

vocabulary

Decision flowchartA step-by-step process: identify the two sides, then pick sin/cos/tan.

Side-pairThe pair of sides (out of hyp, opp, adj) that the problem references.

RearrangeUse multiplication/division to isolate the unknown on one side of the equation.

Mixed problemA practice question where you don't know in advance which trig ratio applies.

VerifyUse Pythagoras (or another check) to make sure your answer is sensible.

Acute angleLess than 90°. All non-right angles in a right triangle are acute.

Spot the Trap

heads-up

Wrong: Picking a ratio at random — gives the wrong answer.

Right: List the two sides involved FIRST, then choose.

Wrong: Picking based on which side is ‘easier to compute’ rather than which fits.

Right: Pick the ratio with EXACTLY the two sides you have/need; the rearrange isolates your unknown.

The Three-Question Test

+5 XP

For any side problem, ask three quick questions:

Q1: What angle is given? → that's $\theta$. Q2: Which side is known (and what name)? Q3: Which side do I want (and what name)? The pair of names tells you which ratio.

angle + known side + wanted side → ratio

$\theta$ first

Always mark $\theta$ on the diagram before labelling other sides.

Name the sides

Calling them ‘opp’, ‘adj’, ‘hyp’ makes the choice automatic.

Sense check

After computing, ask: is the answer the right size compared to the other sides?

Rearrangement Cheat Sheet

+5 XP

Every trig ratio has TWO useful rearrangements — one to find the top, one to find the bottom.

Original	Find top	Find bottom
$\sin\theta = $ opp/hyp	opp = hyp$\cdot \sin\theta$	hyp = opp$/\sin\theta$
$\cos\theta = $ adj/hyp	adj = hyp$\cdot \cos\theta$	hyp = adj$/\cos\theta$
$\tan\theta = $ opp/adj	opp = adj$\cdot \tan\theta$	adj = opp$/\tan\theta$

Top = bottom $\cdot$ ratio ; Bottom = top $/$ ratio

Top from bottom

To find the numerator, multiply the denominator by the trig value.

Bottom from top

To find the denominator, divide the numerator by the trig value.

Memorise pattern

Same rearranging pattern works for all three ratios.

Watch Me Solve It · 3 examples

Watch Me Solve It · Choose sin or cos

+15 XP per step

PROBLEM

In a right triangle, the angle is 50° and the hypotenuse is 14. Find the opposite side (2 d.p.).

1

List sides

Known: hyp = 14, $\theta = 50°$. Want: opp.
2

Pick ratio

opp + hyp → sin (SOH).

sin uses opp/hyp.
3

Compute

opp $= 14\sin 50° \approx 14 \times 0.766 \approx 10.72$

Answer$\approx 10.72$

Watch Me Solve It · Tan is needed

+15 XP per step

PROBLEM

$\theta = 28°$ and the opposite side is 9. Find the adjacent (2 d.p.).

1

List sides

Known: opp = 9, $\theta = 28°$. Want: adj.
2

Pick

opp + adj → tan (TOA).
3

Solve

adj $= 9/\tan 28° \approx 9/0.5317 \approx 16.93$

Answeradj $\approx 16.93$

Watch Me Solve It · Cosine to find hyp

+15 XP per step

PROBLEM

$\theta = 40°$ and the adjacent is 8. Find the hypotenuse (2 d.p.).

1

List sides

adj = 8, $\theta = 40°$. Want: hyp.
2

Pick

adj + hyp → cos (CAH).
3

Solve

hyp $= 8/\cos 40° \approx 8/0.766 \approx 10.44$

Answerhyp $\approx 10.44$

Common Pitfalls

heads-up

Wrong ratio choice

Picking a ratio whose sides don't match the diagram.

Fix: List the two sides BEFORE picking the ratio.

Solving for the wrong side

Computing the side you DIDN'T want.

Fix: Underline the unknown in the problem; solve for that letter explicitly.

Sign / units missing

Reporting a number without units, or rounding too early.

Fix: Keep full calculator precision until the last step; then round AND add units.

Copy Into Your Books

Decision

opp + hyp → sin
adj + hyp → cos
opp + adj → tan

Find top

Multiply bottom by ratio
Find opp = hyp $\sin$
Find adj = hyp $\cos$

Find bottom

Divide top by ratio
Find hyp = opp$/\sin$
Find adj = opp$/\tan$

Always

Sketch
Label
List sides
Pick ratio
Compute

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Ratio Picker

4 problems

Four quick drills to lock in today's skill. Try each, then reveal the answer.

1 Known: opp, $\theta$. Want: hyp. Which ratio?

opp + hyp → sin.sin
2 Known: adj, $\theta$. Want: opp. Which ratio?

opp + adj → tan.tan
3 Known: hyp, $\theta$. Want: adj. Which ratio?

adj + hyp → cos.cos
4 $\theta = 30°$, opp = 6. Find adj (2 d.p.).

adj = $6/\tan 30° \approx 10.39$.$\approx 10.39$

Complete in your workbook.

Quick Check · 5 questions

Given $\theta$, hyp, and want opp. Use:

+10 XP

$\theta = 60°$, adj = 4. Find opp (2 d.p.):

+10 XP

$\theta = 35°$, hyp = 20. Find adj (2 d.p.):

+10 XP

$\theta = 25°$, opp = 5. Find hyp (2 d.p.):

+10 XP

Best first step in ANY trig side problem:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Find the requested side in each case to 2 d.p. (a) $\theta = 50°$, hyp = 9, find opp. (b) $\theta = 25°$, adj = 14, find opp. (c) $\theta = 70°$, opp = 18, find hyp.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. Write a one-sentence rule for deciding between sin, cos and tan in a right-triangle side problem.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A right triangle has angle 32° and hypotenuse 10. Without using Pythagoras, find both legs (2 d.p.). Then verify using Pythagoras.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — SOH.

2. D — $4\tan 60° \approx 6.93$.

3. B — $20\cos 35° \approx 16.38$.

4. C — $5/\sin 25° \approx 11.83$.

5. A — Pair identification first.

Show Your Working Model Answers

Q6 (3 marks): (a) opp = $9\sin 50° \approx 6.89$ [1]. (b) opp = $14\tan 25° \approx 6.53$ [1]. (c) hyp = $18/\sin 70° \approx 19.16$ [1].

Q7 (2 marks): Identify which two of opp, adj, hyp are involved [1]. Then: opp+hyp → sin, adj+hyp → cos, opp+adj → tan [1].

Q8 (4 marks): opp $= 10\sin 32° \approx 5.30$ [1]. adj $= 10\cos 32° \approx 8.48$ [1]. Pythagoras check: $5.30^2 + 8.48^2 \approx 28.09 + 71.91 = 100.00 = 10^2$ [1]. The check confirms the sides are correct — rounding errors are tiny [1].

Stretch Challenge · +25 XP, +10 coins

Without a calculator

Using the special angles 30°, 45°, 60° (where you might know exact values like $\sin 30° = 0.5$, $\cos 60° = 0.5$, $\tan 45° = 1$), find an exact answer for: a right triangle with $\theta = 30°$ and hyp = 12, find opp.

Reveal solution

opp = $12 \cdot \sin 30° = 12 \cdot 0.5 = 6$ exactly (no rounding).

Quick Review

Decision

opp+hyp→sin, adj+hyp→cos, opp+adj→tan

Find top

Top = bottom $\cdot$ ratio

Find bottom

Bottom = top $/$ ratio

Always sketch

Label sides before picking ratio

DEG mode

$\sin 30° = 0.5$ check

Verify

Use Pythagoras as a final check

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