Lesson 7 ~25 min Unit 3 · Trigonometry +85 XP

Sine Ratio — Finding a Side

Use $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$ to calculate either the opposite (opp = hyp $\times$ sin) or the hypotenuse (hyp = opp $\div$ sin) when an angle and one side are known.

Today's hook: A ramp rises at 15° to the horizontal. The ramp itself is 8 m long. The vertical height is the opposite side — how high does the ramp rise?

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Think First

warm-up

You know the hypotenuse (8 m) and the angle (15°), and you want the opposite side. Which side of the equation $\sin 15° = $ opp $/$ 8 do you multiply by 8 to isolate opp?

Record your answer in your workbook.

The Big Idea

+5 XP

When the sides we know and want involve opp and hyp, sine is the tool. We rearrange $\sin\theta = $ opp$/$hyp into TWO useful forms.

To find the opposite: opp = hyp $\times \sin\theta$. To find the hypotenuse: hyp = opp $\div \sin\theta$. Always set up $\sin\theta = $ opp$/$hyp first, then rearrange to isolate the unknown. Round to 2 decimal places unless told otherwise.

opp = hyp $\times \sin\theta$ or hyp = opp $\div \sin\theta$

Set up sin equation

$\sin\theta = $ opp$/$hyp — always start there.

Rearrange

Multiply both sides by hyp to find opp; divide by sin to find hyp.

Calculator in DEG

Set calculator mode to DEG (not RAD) before pressing sin.

What You'll Master

objectives

Know

$\sin\theta = $ opp$/$hyp
Rearranged: opp $=$ hyp$\cdot \sin\theta$ and hyp $=$ opp$/\sin\theta$
Round answers to 2 d.p. unless told otherwise

Understand

Why we use sin when opp and hyp are involved
Why dividing by $\sin\theta$ (less than 1) makes the answer bigger
That the calculator must be in DEG mode for these problems

Can Do

Identify the right ratio (sin) when given opp/hyp pair
Solve for opp or hyp using sine
Use the calculator's sin and $\sin^{-1}$ buttons confidently

Words You Need

vocabulary

$\sin\theta$Sine of $\theta$: ratio opp/hyp for the angle $\theta$ in a right triangle.

DEG modeThe calculator mode for degree measurement of angles — ALWAYS set to DEG for these problems.

Opp sideThe side directly across from the angle $\theta$ in a right triangle.

Multiply both sidesTo isolate opp in $\sin\theta = $ opp/hyp, multiply each side by hyp.

Divide to isolate hypFrom opp $= $ hyp $\cdot \sin\theta$, divide both sides by $\sin\theta$ to get hyp.

Round (2 d.p.)Round to two digits after the decimal point. Standard precision for trig answers.

Spot the Trap

heads-up

Wrong: “opp = hyp + $\sin\theta$.” No — it's multiplication: opp = hyp $\times \sin\theta$.

Right: Always: opp = hyp $\times \sin\theta$. The sine is a multiplier, not an addition.

Wrong: Using radians (RAD mode) by accident gives wildly wrong answers.

Right: Check the calculator screen: it should say DEG (or D) before computing.

When to use Sine

+5 XP

Choose sin if and only if the two sides involved (the known and the unknown) are the opposite and the hypotenuse.

Look at the diagram. If you know opp + angle, and want hyp: rearrange sin to hyp = opp/$\sin\theta$. If you know hyp + angle, and want opp: rearrange to opp = hyp $\times \sin\theta$. Tip: sin appears whenever the diagram has the side across from $\theta$ AND the long slanted side.

sin needed when problem mentions opp and hyp

Identify sides first

Label opp, adj, hyp before choosing the ratio.

Across + slant

If across-from-$\theta$ and the slant side are involved, use sin.

Hyp on bottom

$\sin\theta = $ opp/hyp — remember hyp goes underneath.

Calculator Setup

+5 XP

Trigonometric calculation goes wrong if the calculator is in radians. Always check.

Step	Key sequence
Set DEG mode	SHIFT → MODE/SETUP → pick DEG (Casio: usually MODE 1)
Compute $\sin 30°$	sin 3 0 = → 0.5
opp = 10 $\times \sin 35°$	1 0 $\times$ sin 3 5 = → 5.74

DEG mode — check before every trig calculation

DEG D

Screen should show D or DEG, not R or RAD.

$\sin 30° = 0.5$

Use this as a quick sanity check that you're in DEG.

Close brackets

sin(35) needs the right bracket on some calculators.

Watch Me Solve It · 3 examples

Watch Me Solve It · Ramp height

+15 XP per step

PROBLEM

A ramp 8 m long rises at 15° to the ground. Find the vertical height it reaches (2 d.p.).

1

Identify sides

hyp = 8 (ramp), opp = height, $\theta = 15°$
2

Set up sin

$\sin 15° = $ opp$/8$
3

Solve

opp $= 8 \sin 15° \approx 8 \times 0.2588 \approx 2.07$ m

AnswerHeight $\approx 2.07$ m

Watch Me Solve It · Find the hypotenuse

+15 XP per step

PROBLEM

A right triangle has an angle of 28° with the opposite side 5 cm. Find the hypotenuse.

1

Set up

$\sin 28° = 5/$hyp
2

Rearrange

hyp $= 5/\sin 28°$
3

Compute

hyp $\approx 5/0.4695 \approx 10.65$ cm

Answerhyp $\approx 10.65$ cm

Watch Me Solve It · Climbing a slope

+15 XP per step

PROBLEM

A walking trail rises at 12°. After 200 m along the trail, how much higher are you (2 d.p.)?

1

Identify

hyp = 200 m (trail), opp = vertical rise, $\theta = 12°$
2

Apply sin

rise = 200 $\sin 12°$
3

Compute

$\approx 200 \times 0.2079 \approx 41.58$ m

Answer$\approx 41.58$ m higher

Common Pitfalls

heads-up

Wrong calculator mode

Computing in RAD mode gives nonsense like $\sin 30 \approx -0.99$.

Fix: Set DEG mode and verify $\sin 30° = 0.5$ before solving.

Wrong rearrangement

Writing opp = hyp / sin instead of opp = hyp $\times$ sin.

Fix: From sin = opp/hyp: multiply both sides by hyp → opp = hyp $\cdot$ sin.

Forgetting units

Writing ‘5.74’ instead of ‘5.74 m’.

Fix: Always include the unit on the final answer.

Copy Into Your Books

Sine ratio

$\sin\theta = $ opp/hyp
opp = hyp $\times \sin\theta$
hyp = opp $/ \sin\theta$

When to use

Sides involved: opp + hyp
Includes angle $\theta$
Across-from-$\theta$ + slant

Calculator

DEG mode required
Check: $\sin 30° = 0.5$
Use $\sin$ key

Method

Label sides
Set up sin
Rearrange
Compute, 2 d.p.

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Sine Side Hunt

4 problems

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

1 opp = 6, $\theta = 35°$. Find hyp (2 d.p.).

hyp = $6/\sin 35° \approx 6/0.5736 \approx 10.46$.$\approx 10.46$
2 hyp = 12, $\theta = 50°$. Find opp (2 d.p.).

opp = $12\sin 50° \approx 12 \times 0.766 \approx 9.19$.$\approx 9.19$
3 hyp = 25, $\theta = 30°$. Find opp.

opp $= 25 \times 0.5 = 12.5$.$= 12.5$
4 opp = 4, $\theta = 60°$. Find hyp (2 d.p.).

hyp $= 4/\sin 60° \approx 4/0.866 \approx 4.62$.$\approx 4.62$

Complete in your workbook.

Quick Check · 5 questions

A ramp of length 10 m rises at 20°. Vertical height (2 d.p.)?

+10 XP

Given $\sin\theta = $ opp/hyp, to find opp:

+10 XP

opp = 8, $\theta = 30°$. Hyp?

+10 XP

To check your calculator is in DEG mode, compute $\sin 30°$. You should get:

+10 XP

A 20 m kite string makes an angle of 55° with the ground. How high above the ground is the kite (2 d.p.)?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. For each right triangle, find the requested side to 2 d.p. (a) hyp = 14, $\theta = 25°$, find opp. (b) opp = 7, $\theta = 40°$, find hyp. (c) hyp = 9, $\theta = 60°$, find opp.

Answer in your workbook.

ApplyEasy2 MARKS

Q7. A ramp 8 m long rises at an angle of 15° from the horizontal. Find the vertical height it reaches.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A water-slide is built at 38° to the horizontal. The slide's vertical drop is 12 m. (a) Find the length of the slide. (b) If the angle is reduced to 25° with the same vertical drop, find the new slide length. (c) Comment on what happens to the slide length as the angle decreases.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $10\sin 20° \approx 3.42$ m.

2. A — Multiply both sides by hyp.

3. D — hyp = $8/0.5 = 16$.

4. A — $\sin 30° = 0.5$.

5. C — $20\sin 55° \approx 16.38$.

Show Your Working Model Answers

Q6 (3 marks): (a) opp $= 14\sin 25° \approx 5.92$ [1]. (b) hyp $= 7/\sin 40° \approx 10.89$ [1]. (c) opp $= 9\sin 60° \approx 7.79$ [1].

Q7 (2 marks): opp $= 8 \sin 15°$ [1] $\approx 2.07$ m [1].

Q8 (4 marks): (a) hyp $= 12/\sin 38° \approx 19.49$ m [1]. (b) hyp $= 12/\sin 25° \approx 28.39$ m [1]. (c) Slide length increases as the angle decreases [1]. Reason: smaller $\sin\theta$ in the denominator means larger hypotenuse for the same opp [1].

Stretch Challenge · +25 XP, +10 coins

Two-stage slide

A water slide drops 6 m over its first section at 30°, then drops another 4 m over a second section at 50°. Find the total length of the slide.

Reveal solution

Section 1: hyp$_1 = 6/\sin 30° = 12$ m. Section 2: hyp$_2 = 4/\sin 50° \approx 5.22$ m. Total $\approx 17.22$ m.

Quick Review

Sine ratio

$\sin\theta = $ opp/hyp

Find opp

opp $=$ hyp $\cdot \sin\theta$

Find hyp

hyp $=$ opp $/ \sin\theta$

Calculator

DEG mode — check $\sin 30° = 0.5$

Round

2 d.p. by default

When to use

opp + hyp involved

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