Lesson 2 ~35 min Unit 3 · Trigonometry +85 XP

Finding Unknown Sides

Learn to select the correct trig ratio and use your calculator to find missing sides in right-angled triangles. From building ramps to calculating cliff heights, this skill unlocks real-world measurement.

Today's hook: A roof truss forms a right-angled triangle. If you know one acute angle and the hypotenuse, you can calculate every other side without ever climbing up to measure.

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Think First

warm-up

A roof truss forms a right-angled triangle. If you know one of the acute angles and the length of the hypotenuse, what else could you work out? How?

Record your answer in your workbook.

The Big Idea

+5 XP to read

Every side-finding problem follows the same three-step pattern. Master these three steps and you can solve any right-angled triangle problem -- from a wheelchair ramp to the height of a cliff.

Label the sides relative to the marked angle. Choose the ratio that connects the side you know to the side you want. Solve the equation, rearranging if the unknown is on the bottom.

$\text{Opposite} = \text{Hypotenuse} \times \sin \theta$

Label first

Name O, A and H before you write any equation.

Match the pair

Given side + unknown side = your ratio choice.

Make it the subject

Rearrange so the unknown stands alone before calculating.

What You'll Master

objectives

Know

How to rearrange $\sin \theta$, $\cos \theta$ and $\tan \theta$ to make any side the subject
That your calculator must be in degree (DEG) mode
How to round to a given number of decimal places

Understand

That choosing the correct ratio depends only on which two sides are involved
Why the answer is always a length and must carry correct units

Can Do

Label the opposite, adjacent and hypotenuse for the marked angle
Select sin, cos or tan based on the given information
Write and solve an equation to find the unknown side

Words You Need

vocabulary

Unknown sideThe side whose length you need to calculate.

RearrangingAlgebraically changing a formula so the quantity you want is on its own.

Degree mode (DEG)The calculator setting that treats angles as degrees, not radians.

Decimal placesThe number of digits after the decimal point in a rounded answer.

SubjectThe variable standing alone on the left-hand side of an equation.

Significant figuresThe digits in a number that carry meaningful information about its precision.

Spot the Trap

heads-up

Wrong: Using $\sin \theta = \frac{\text{hypotenuse}}{\text{opposite}}$ to find the opposite side.

Right: Always write the ratio correctly first: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$, then rearrange to $\text{Opposite} = \text{Hypotenuse} \times \sin \theta$.

Wrong: Leaving the calculator in RAD mode and getting an answer like $0.0349$ instead of $2$.

Right: Check the top-right corner of your calculator screen for DEG before you start.

Parts of the Whole

+5 XP

Every side-finding problem has the same three ingredients: a marked angle, a given side whose length you know, and an unknown side you need to find.

Before you touch your calculator, identify the marked angle, the given side and the unknown side. These three pieces determine everything else.

$\text{Opposite} = \text{Hypotenuse} \cdot \sin \theta$

Identify the angle first

Everything else is defined relative to this angle.

Spot the given side

The number in the question is your starting point.

Name the unknown

Call it $x$ so you have a clear target to solve for.

Match the Pair

+5 XP

The ratio you choose depends only on which two sides are involved. Opposite + Hypotenuse = sine. Adjacent + Hypotenuse = cosine. Opposite + Adjacent = tangent.

Look at the given side and the unknown side. Match that pair to one of the three ratios. If the hypotenuse is involved, you are using sine or cosine. If not, use tangent.

$\sin \theta = \frac{O}{H}$

O + H = sine

If the hypotenuse is involved with opposite, use sin.

A + H = cosine

If the hypotenuse is involved with adjacent, use cos.

O + A = tangent

If the hypotenuse is NOT involved, use tan.

Make It the Subject

+5 XP

When the unknown side is on the bottom of the fraction, multiply both sides by it. Then isolate. Think of it as solving a mini-equation.

If the unknown is on top, multiply. If it is on the bottom, multiply both sides by the unknown first, then divide. Never reach for the calculator until the unknown stands alone.

$x = \frac{15}{\tan 50°}$

Unknown on top = multiply

$\sin \theta = \frac{x}{12}$ becomes $x = 12 \times \sin \theta$.

Unknown on bottom = divide

$\tan \theta = \frac{15}{x}$ becomes $x = \frac{15}{\tan \theta}$.

Write every step

Examiners award marks for correct rearrangement.

Check and Round

+5 XP

Before you press sine, check the screen says DEG. After you calculate, round to the requested precision and always attach the correct unit.

CASIO: Press SHIFT then SETUP, select Angle Unit, then Degree. Look for D at the top. TI: Press MODE, scroll to Degree, press ENTER. If you see R or G, your answer will be completely wrong.

$7.166 \rightarrow 7.2$ m

DEG not RAD

RAD mode gives answers about 57 times too small.

Full display then round

Write the unrounded value, then state the rounded answer.

Units are part of the answer

$7.2$ is a number. $7.2$ m is a length.

Watch Me Solve It · 3 examples

Watch Me Solve It · Finding the opposite

+15 XP per step

PROBLEM

A right-angled triangle has θ = 35° and hypotenuse = 12 cm. Find the length of the side opposite θ to 1 decimal place.

1

Label the sides

Hypotenuse = 12 cm, Opposite = $x$, Adjacent = unknown (not needed)

Identify which sides are involved before choosing a ratio.
2

Choose the ratio

We have Hypotenuse and want Opposite → use $\sin \theta$

Opposite + Hypotenuse pairs with sine (SOH).
3

Write the equation

$\sin 35° = \frac{x}{12}$
4

Solve and round

$x = 12 \times \sin 35° = 12 \times 0.5736 = 6.883$ → 6.9 cm

Check DEG mode, calculate, then round to 1 decimal place.

Answer$6.9$ cm

Watch Me Solve It · Finding the adjacent

+15 XP per step

PROBLEM

A right-angled triangle has θ = 50° and opposite side = 15 m. Find the adjacent side to 2 decimal places.

1

Label the sides

Opposite = 15 m, Adjacent = $x$, Angle = 50°
2

Choose the ratio

We have Opposite and want Adjacent → use $\tan \theta$

Opposite + Adjacent pairs with tangent (TOA).
3

Write and rearrange

$\tan 50° = \frac{15}{x}$ → $x = \frac{15}{\tan 50°}$

The unknown is on the bottom -- multiply both sides by $x$ first, then divide.
4

Calculate and round

$x = \frac{15}{1.1918} = 12.586$ → 12.59 m

Answer$12.59$ m

Watch Me Solve It · Wheelchair ramp

+15 XP per step

PROBLEM

A wheelchair ramp must rise 0.5 m at an angle of 4° to the horizontal. Find the length of the ramp surface (hypotenuse) to 1 decimal place.

1

Label the sides

Opposite = 0.5 m (rise), Hypotenuse = $x$ (ramp), Angle = 4°
2

Choose the ratio

We have Opposite and want Hypotenuse → use $\sin \theta$
3

Write and rearrange

$\sin 4° = \frac{0.5}{x}$ → $x = \frac{0.5}{\sin 4°}$
4

Calculate and round

$x = \frac{0.5}{0.06976} = 7.166$ → 7.2 m

A shallow angle means a long ramp -- the answer makes sense.

Answer$7.2$ m

Common Pitfalls

heads-up

Choosing the wrong ratio

Given adjacent and hypotenuse but using tan because "it feels right". Always match the pair of sides to SOH / CAH / TOA before writing anything.

Fix: say aloud "I have Opposite and Hypotenuse -- that is sine" before opening your calculator.

Calculator in RAD mode

A calculator in RAD mode makes $\sin 30° = 0.5$ into something like $-0.988$. Every answer is completely wrong.

Fix: look for D or DEG in the top corner of the screen before every lesson.

Forgetting units

Writing "7.2" instead of "7.2 m" loses easy marks. The unit is part of the answer.

Fix: after every final number, ask yourself "7.2 what?" and write the unit.

Copy Into Your Books

Rearranged Formulas

$\text{Opposite} = \text{Hypotenuse} \times \sin \theta$
$\text{Adjacent} = \text{Hypotenuse} \times \cos \theta$
$\text{Opposite} = \text{Adjacent} \times \tan \theta$

Hypotenuse Formulas

$\text{Hypotenuse} = \frac{\text{Opposite}}{\sin \theta}$
$\text{Hypotenuse} = \frac{\text{Adjacent}}{\cos \theta}$
$\text{Adjacent} = \frac{\text{Opposite}}{\tan \theta}$

The Three Steps

1. Label -- name O, A, H
2. Choose -- match pair to ratio
3. Solve -- rearrange and calculate

Calculator Check

Confirm DEG mode
Write full display before rounding
Always include units

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems finding different unknown sides. Work each one, then reveal the answer.

1 θ = 28°, hypotenuse = 18 cm. Find the opposite side to 1 decimal place.

$\sin 28° = \frac{x}{18}$ → $x = 18 \times \sin 28° = 18 \times 0.4695$$= 8.5$ cm
2 A kite string makes 52° with the horizontal. The kite is 35 m above ground. How long is the string to the nearest metre?

$\sin 52° = \frac{35}{x}$ → $x = \frac{35}{\sin 52°} = \frac{35}{0.7880}$$= 44$ m
3 θ = 60°, adjacent = 8 cm. Find the opposite side to 1 decimal place.

$\tan 60° = \frac{x}{8}$ → $x = 8 \times \tan 60° = 8 \times 1.732$$= 13.9$ cm
4 θ = 40°, opposite = 10 m. Find the hypotenuse to 1 decimal place.

$\sin 40° = \frac{10}{x}$ → $x = \frac{10}{\sin 40°} = \frac{10}{0.6428}$$= 15.6$ m

Complete in your workbook.

Quick Check · 5 questions

θ = 40°, hypotenuse = 10 cm. Which ratio finds the opposite?

+10 XP

θ = 25°, adjacent = 8 m. The opposite side is found using:

+10 XP

Which calculator check should you perform first?

+10 XP

If $\cos 60° = \frac{x}{14}$, the value of $x$ is:

+10 XP

A flagpole casts a shadow. Angle of elevation = 35° and shadow (adjacent) = 12 m. The height is:

+10 XP

Show Your Working · 3 questions

Show Your Working

7 marks total

Apply Easy 2 MARKS

Q6. A right-angled triangle has angle θ = 42° and hypotenuse 20 cm. (a) Find the opposite side to 1 decimal place. (b) Find the adjacent side to 1 decimal place.

Answer in your workbook.

Apply Medium 2 MARKS

Q7. A surveyor stands 45 m from the base of a cliff. The angle of elevation to the top is 38°. (a) Draw a diagram and label the known and unknown sides. (b) Choose the correct ratio and calculate the height of the cliff to the nearest metre.

Answer in your workbook.

Analyse Hard 3 MARKS

Q8. A kayak ramp rises at 6° to the horizontal. The vertical rise is 0.8 m. (a) Find the length of the ramp surface to 2 decimal places. (b) Find the horizontal distance to 2 decimal places. (c) If the vertical rise is doubled but the angle stays the same, explain what happens to the ramp length and horizontal distance.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B -- Opposite + Hypotenuse = sine.

2. C -- Opposite + Adjacent = tangent.

3. B -- Always check DEG mode first.

4. B -- $x = 14 \times \cos 60° = 14 \times 0.5 = 7$.

5. C -- Height = opposite, shadow = adjacent. Use tan.

Show Your Working Model Answers

Q6 (2 marks): (a) $\sin 42° = \frac{x}{20}$ → $x = 20 \times \sin 42° = 13.4$ cm [1]. (b) $\cos 42° = \frac{y}{20}$ → $y = 20 \times \cos 42° = 14.9$ cm [1].

Q7 (2 marks): (a) Diagram showing right triangle with base 45 m, height $h$, angle 38° [1]. (b) $\tan 38° = \frac{h}{45}$ → $h = 45 \times \tan 38° = 45 \times 0.7813 = 35$ m [1].

Q8 (3 marks): (a) $\sin 6° = \frac{0.8}{L}$ → $L = \frac{0.8}{\sin 6°} = 7.65$ m [1]. (b) $\tan 6° = \frac{0.8}{d}$ → $d = \frac{0.8}{\tan 6°} = 7.61$ m [1]. (c) Doubling the opposite while keeping the angle constant doubles both the hypotenuse and adjacent because the triangle is scaled by factor 2 [1].

Stretch Challenge · +25 XP, +10 coins

The Park Path

A rectangular park has a diagonal path 120 m long that makes an angle of 35° with the longer side. Find the length and width of the park to 1 decimal place. Verify your answers using Pythagoras' theorem.

Reveal solution

Length = $120 \times \cos 35° = 120 \times 0.8192 = 98.3$ m. Width = $120 \times \sin 35° = 120 \times 0.5736 = 68.8$ m. Verification: $98.3^2 + 68.8^2 = 9662.9 + 4733.4 = 14396.3 \approx 14400 = 120^2$.

Quick Review

Label first

Name O, A and H

Match the pair

O+H=sin, A+H=cos, O+A=tan

Make it the subject

Isolate the unknown before calculating

DEG mode

Check before every problem

Round carefully

Full display, then round

Always use units

cm, m or km

Interactive: Side Finder

Practise choosing the correct ratio and calculating missing sides on random triangles.

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DAILY CHALLENGE · TODAY

The Side-Finder Sprint

Find the opposite side when θ = 50° and hypotenuse = 12 cm, then find the adjacent side, both to 1 decimal place, in under 60 seconds for double coins!

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Lesson 3 -- Finding Unknown Angles

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