Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute Triangles

Using Trigonometric Ratios to Find Side Lengths

2.2 2.3About 70 minutes

Learning goals

I can choose the correct trigonometric ratio from the sides I know and the side I want.
I can solve for an unknown side when it is in the numerator or the denominator.
I can round correctly and check that my answer is reasonable.

Words to know

solve a triangle
Find unknown sides or angles using the information given.

numerator / denominator
The top / bottom of a fraction. Where the unknown sits decides whether you multiply or divide.

angle of inclination
The angle a sloped object (ladder, ramp, roof) makes with the horizontal ground.

From a ratio to a length

Last lesson, sin, cos and tan were ratios we calculated from sides. Now we turn it around. If you know one angle and one side of a right triangle, the calculator gives you the ratio, and a short equation gives you any other side. This is how a carpenter finds the height of a roof or a firefighter works out how high a ladder reaches.

The five steps for finding a side. Step 1 always comes first: label the sides from the given angle.

Which ratio?

Look at the two sides involved: the one you know and the one you want.

opposite and hypotenuse → sin
adjacent and hypotenuse → cos
opposite and adjacent → tan

The third side is not used at all.

Left (Example 1): the unknown x is opposite 40° and the hypotenuse 12 is known, so use sin. Right (Example 3): 5 is opposite 32° and the unknown x is the hypotenuse, so use sin again, but x ends up on the bottom.

Worked example

Example 1: Unknown on top (sin)

In △ABC, ∠C = 90°, ∠A = 40° and AB = 12 cm. Find BC (called x).

From ∠A: AB is the hypotenuse (12), BC is opposite (x).
Opposite and hypotenuse → sin. sin 40° = x/12.
Multiply both sides by 12: x = 12 sin 40°.
Calculator: 12 × sin 40° = 7.713... ≈ 7.7.

BC ≈ 7.7 cm

Worked example

Example 2: Unknown on top (tan)

A right triangle has an angle of 25°. The side adjacent to it is 8 m. Find the opposite side.

Opposite and adjacent → tan. tan 25° = x/8.
x = 8 tan 25° = 3.730... ≈ 3.7.

The opposite side is about 3.7 m.

Worked example

Example 3: Unknown on the bottom

In △DEF, ∠F = 90°, ∠D = 32° and EF = 5 cm. Find DE (called x).

From ∠D: EF is opposite (5), DE is the hypotenuse (x).
sin 32° = 5/x.
Multiply both sides by x: x sin 32° = 5. Divide by sin 32°: x = 5 ÷ sin 32°.
x = 5 ÷ 0.5299... = 9.435... ≈ 9.4.
Check: the hypotenuse (9.4) is longer than the leg (5). ✓

DE ≈ 9.4 cm

Worked example

Example 4: Unknown on the bottom (tan)

A right triangle has a 50° angle. The side opposite it is 9 cm. Find the adjacent side.

Opposite and adjacent → tan 50° = 9/x.
x = 9 ÷ tan 50° = 9 ÷ 1.1917... = 7.551... ≈ 7.6.
Check: the angle is more than 45°, so the opposite side should be longer than the adjacent side. 9 > 7.6. ✓

The adjacent side is about 7.6 cm.

Rounding and calculator tips

Type the whole expression at once (for example 9 ÷ tan(50)) so you never round in the middle.
Round only the final answer: usually to one decimal place for lengths, unless the question says otherwise.
Quick check: the hypotenuse must be the longest side. If your "hypotenuse" comes out shorter than a leg, you multiplied where you should have divided.

Worked example

Example 5: How high does the ladder reach?

A 5.0 m ladder leans against a wall. It makes an angle of 70° with the level ground. How high up the wall does it reach? How far is its foot from the wall?

The ladder is the hypotenuse (5.0 m). The height h is opposite the 70° angle; the distance d from the wall is adjacent.
sin 70° = h/5.0, so h = 5.0 sin 70° = 4.698... ≈ 4.7 m.
cos 70° = d/5.0, so d = 5.0 cos 70° = 1.710... ≈ 1.7 m.

The ladder reaches about 4.7 m up the wall, and its foot is about 1.7 m from the wall.

The ladder, the wall and the ground make a right triangle. The ladder is the hypotenuse, and the height on the wall is opposite the 70° angle at the ground.

Just for fun

Why did the ladder go to trigonometry class?

It wanted to know how high it could reach in life, and to finally learn its angle.

Hands-on activity

How high does the broom reach?

You need: a broom or mop, a measuring tape, a protractor, and a wall with a non-slip floor. A grown-up should hold the broom steady.

Measure the length of the broom handle from the floor end to the top end.
Lean it against the wall. Hold the protractor at the floor end, against the floor, and measure the angle between the floor and the broom.
Use sin to predict how high the top of the broom touches the wall. Use cos to predict how far the bottom is from the wall.
Now measure both. How close were your predictions?
Change the angle and repeat.

What to look for: predictions within a few centimetres. If the angle is 60° and the broom is 1.3 m, the predicted height is 1.3 sin 60° ≈ 1.13 m and the distance is 1.3 cos 60° = 0.65 m. Larger errors usually come from reading the protractor.

Your turn

Hypotenuse 15 cm, angle 28°. Find the side opposite the angle.

Hypotenuse 20 m, angle 62°. Find the side adjacent to the angle.

Adjacent side 11 cm, angle 36°. Find the opposite side.

Opposite side 7 cm, angle 41°. Find the hypotenuse.

Adjacent side 13 m, angle 55°. Find the hypotenuse.

Opposite side 6 cm, angle 22°. Find the adjacent side.

A kite string is 60 m long and makes a 48° angle with the ground. Ignoring the height of the person's hand, how high is the kite?

More practice: warm-up

You know the hypotenuse and want the side opposite the angle. Which ratio do you use?

Solve x/7 = 0.5.

Solve 12/x = 0.6.

Evaluate 10 cos 30° to one decimal place.

You know the angle and the adjacent side, and you want the hypotenuse. Which ratio?

More practice: on your own

In △ABC, ∠C = 90°, ∠A = 34° and AB = 18 cm. Find BC and AC.

In △ABC, ∠C = 90°, ∠B = 67° and BC = 4.5 m. Find AC.

A ramp makes an angle of 4.8° with the ground and rises 0.75 m. How long is the sloped surface of the ramp?

A playground slide is 3.2 m long and makes a 38° angle with the ground. How high is the top of the slide?

Ravi's work: "Opposite = 9, angle 50°, find adjacent x: x = 9 tan 50° ≈ 10.7." What went wrong? Find the correct answer.

Challenge

An isosceles triangle has two equal sides of 10 cm and base angles of 70°. Find its height and its base.

A right triangle has a hypotenuse of 25 cm and a 30° angle. Find its perimeter.

The diagonal of a rectangle is 15 cm and makes a 27° angle with the longer side. Find the area of the rectangle.

Check your understanding

Quiz

A right triangle has a hypotenuse of 10 and a 30° angle. How long is the side opposite the 30° angle?

x is adjacent to a 40° angle and the hypotenuse is 8. Which equation is correct?

The side opposite a 35° angle is 6 cm. What is the hypotenuse?

In sin 32° = 5/x, the unknown is in the denominator. What do you do?

A 4 m ladder makes a 75° angle with the ground. About how high does it reach?

The side adjacent to a 60° angle is 10 cm. How long is the opposite side?

For parents and teachers

Big idea: Expectations 2.2 and 2.3: finding side lengths of right triangles with the primary trigonometric ratios, including simple real-life uses such as ladders and ramps.

How to help: Ask your teen to say the five steps out loud for one question: label, circle, choose, write, solve. The broom activity gives instant feedback on whether the answer makes sense. Keep the household broom steady for safety.

Common difficulties:

Multiplying when the unknown is in the denominator (and the reverse). The "hypotenuse must be longest" check catches many of these.
Rounding the ratio early (using 0.53 instead of sin 32°). Type the full expression into the calculator.
Choosing the ratio by guessing. The two sides named in the question decide the ratio.