C. Trigonometric Functions · Trigonometry of Acute Triangles

Right-Triangle Trigonometry: Finding Side Lengths

C1.1About 70 minutes

Learning goals

I can name the hypotenuse, opposite side and adjacent side for a given acute angle in a right triangle.
I can choose sine, cosine or tangent to find an unknown side.
I can solve the equation correctly whether the unknown side is in the numerator or the denominator.
I can check a side length with the Pythagorean theorem and with "bigger angle, longer opposite side".

Words to know

hypotenuse
The longest side of a right triangle, across from the 90° angle.

opposite side
The leg across from the angle you are working with.

adjacent side
The leg that touches the angle you are working with (and is not the hypotenuse).

primary trigonometric ratios
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Pythagorean theorem
In a right triangle with legs a and b and hypotenuse c, a² + b² = c².

Measuring without a tape measure

It is February, there is snow on the roof, and someone wants to know if a 6 m ladder will reach the eavestrough. Nobody wants to climb up with a tape measure first. With one angle and one length, trigonometry gives the answer from the ground.

Remember from Grade 10: in Principles of Mathematics (MPM2D) you met the three primary trigonometric ratios, sine, cosine and tangent, and used them to find sides and angles of right triangles. This unit starts by sharpening those skills, then moves to triangles with no right angle using the sine law and the cosine law. Keep your calculator in degree mode (check: sin 30° should give 0.5).

Naming the sides

In a right triangle, the names of the two legs depend on which acute angle you are looking from.

The hypotenuse is always across from the right angle. It is the longest side.
The opposite side is across from your angle θ.
The adjacent side touches θ and is not the hypotenuse.

If you switch to the other acute angle, the opposite and adjacent sides trade names. The hypotenuse never changes.

The same right triangle twice. When θ is at A (left), BC is opposite and AC is adjacent. When θ is at B (right), they swap. The hypotenuse AB stays the same.

The primary ratios (SOH CAH TOA)

For an acute angle θ in a right triangle:

Sin θ = Opposite ÷ Hypotenuse
Cos θ = Adjacent ÷ Hypotenuse
Tan θ = Opposite ÷ Adjacent

The ratios depend only on the angle, not on the size of the triangle, because all right triangles with the same acute angle are similar.

A four-step routine

Sketch and label the angle you know, the side you know and the side you want.
Name those two sides (opposite, adjacent or hypotenuse) from the angle.
Choose the ratio that uses exactly those two sides.
Solve the equation, then round at the end (lengths to one decimal place unless told otherwise).

Watch where the unknown lands. If it is on top of the fraction, multiply. If it is on the bottom, the unknown and the trig value swap places: divide.

Two kinds of equations. Left: the unknown is the numerator, so multiply. Right: the unknown is the denominator, so divide the known side by the trig value.

Worked example

Example 1: Will the ladder reach?

A 6.0 m ladder leans against a house and makes a 74° angle with the level ground. How high up the wall does it reach, and how far is its foot from the wall?

Sketch: the ladder is the hypotenuse (6.0 m). The 74° angle is at the foot of the ladder, on the ground.
Height h is opposite the 74° angle. Opposite and hypotenuse → sine.
sin 74° = h / 6.0, so h = 6.0 sin 74° = 5.768... ≈ 5.8 m.
Distance d from the wall is adjacent to 74°. Adjacent and hypotenuse → cosine.
d = 6.0 cos 74° = 1.654... ≈ 1.7 m.
Check with Pythagoras: 5.768² + 1.654² = 36.00 = 6.0². ✓

The ladder reaches about 5.8 m up the wall, with its foot about 1.7 m from the wall.

Example 1. The ladder is the hypotenuse. The height is opposite the 74° angle (sine) and the distance along the ground is adjacent to it (cosine).

Worked example

Example 2: Unknown in the denominator

A wheelchair ramp at a community centre must rise 0.45 m. The ramp makes an angle of 4.8° with the ground. How long is the sloped surface of the ramp? How much horizontal space does the ramp take up?

The rise (0.45 m) is opposite 4.8°. The sloped surface is the hypotenuse x. Use sine.
sin 4.8° = 0.45 / x. The unknown is on the bottom.
Multiply both sides by x: x sin 4.8° = 0.45. Divide by sin 4.8°: x = 0.45 ÷ sin 4.8°.
x = 0.45 ÷ 0.0837... = 5.378... ≈ 5.38 m.
Horizontal run y is adjacent; use tangent: tan 4.8° = 0.45 / y, so y = 0.45 ÷ tan 4.8° ≈ 5.36 m.
Check: the hypotenuse should be just a little longer than the run, because the angle is small. ✓

The ramp surface is about 5.38 m long and takes up about 5.36 m of horizontal space.

Worked example

Example 3: Tangent and a shadow

On a sunny winter afternoon, a maple tree casts a 14 m shadow on flat ground. The sun's rays make a 38° angle with the ground. How tall is the tree?

The tree and its shadow form the right angle. The 38° angle is at the tip of the shadow.
Height h is opposite 38°. The shadow is adjacent. Use tangent.
tan 38° = h / 14, so h = 14 tan 38° = 10.938... ≈ 10.9 m.

The tree is about 10.9 m tall.

Quick checks

The hypotenuse must be the longest side.
In a right triangle, the larger acute angle is across from the longer leg.
If both legs and the hypotenuse are known, a² + b² should equal c² (up to rounding).
Keep full calculator values until the final answer. Rounding early can change the last digit.

Just for fun

Why did the student refuse to argue with the right triangle?

Because it was always right, and it had the angles to prove it.

Hands-on activity

Broom-handle trigonometry

You need: a broom or mop, a wall, a tape measure, and a protractor (or a free angle-measuring app on a phone).

Measure the length of the broom handle in metres.
Lean it against a wall. Measure the angle between the handle and the floor.
Predict how high up the wall the top touches (sine) and how far the bottom is from the wall (cosine).
Measure both with the tape measure. Find the percentage error: (measured − predicted) ÷ measured × 100%.
Repeat with a steeper angle.

What to look for: predictions within about 5% of the measurements. Bigger errors usually come from reading the protractor, or from measuring to the wrong point (measure to where the handle touches, not to its end if it overhangs).

Your turn

In right △ABC with ∠C = 90°, ∠A = 35° and the hypotenuse AB = 20 cm, find BC.

In right △ABC with ∠C = 90°, ∠A = 35° and AB = 20 cm, find AC.

In right △PQR with ∠Q = 90°, ∠P = 52° and PQ = 8.5 m, find QR.

A right triangle has an angle of 28° and the side opposite it is 9.0 cm. Find the hypotenuse.

A right triangle has an angle of 63° and the side adjacent to it is 4.2 m. Find the opposite side and the hypotenuse.

A kite string is 45 m long and makes a 57° angle with the ground. Ignoring sag and the height of the hand holding it, how high is the kite?

A ski hill run drops 120 m vertically. Its slope makes a 16° angle with the horizontal. How long is the run along the slope?

More practice: warm-up

Which side of a right triangle is always the longest?

Write the ratio for cos θ.

Evaluate sin 40° to four decimal places.

Solve x/12 = tan 25°.

Solve 15/x = cos 50°.

More practice: on your own

A flagpole casts a 9.6 m shadow when the sun is 41° above the horizon. How tall is the flagpole?

A zipline cable is 80 m long and drops at 12° below the horizontal. How far does a rider drop, and how far do they travel horizontally?

A student wrote: "sin 30° = 7/x, so x = 7 sin 30° = 3.5." Find and fix the error.

In right △XYZ, ∠Z = 90°, ∠X = 47° and YZ = 13.0 cm. Find XY and XZ.

A roof rafter makes a 26° angle with the horizontal ceiling joist. The joist spans 4.5 m from the wall to directly below the peak. How long is the rafter?

Challenge

Show that for any acute angle θ, sin θ ÷ cos θ = tan θ.

The instructions for a 3.0 m ladder say it must make an angle between 70° and 76° with the ground. What range of distances from the wall can its foot be?

An isosceles triangle has two equal sides of 10 cm and a base angle of 65°. Find the length of the base and the height, using a right triangle.

Check your understanding

Quiz

In right △ABC with ∠C = 90°, which side is adjacent to ∠A?

Which ratio connects the opposite side and the hypotenuse?

A right triangle has hypotenuse 20 and an angle of 35°. What is the side adjacent to 35°?

sin 28° = 9/x. What is x?

A tree casts a 14 m shadow when the sun is 38° above the ground. Which equation gives the tree's height h?

Your calculator says sin 30° = −0.988. What is wrong?

For parents and teachers

Big idea: Expectation C1.1 asks students to solve problems, including real-world ones such as surveying and navigation, by finding the sides and angles of right triangles with the primary trigonometric ratios. This lesson covers sides; the next one covers angles.

How to help: Ask your teen to say out loud which angle they are looking from, then point to the opposite, adjacent and hypotenuse. Most errors happen before any calculating starts.

Common difficulties:

Calling the wrong leg "opposite" when the angle is at the top of the triangle.
Multiplying when the unknown is in the denominator (Example 2).
Calculator in radian mode: sin 30° must give 0.5.
Rounding in the middle of a problem, then carrying the rounded value forward.