Solve right-triangle problems with primary trigonometric ratios

Ontario Grade 11 Mathematics

Trigonometric Functions

Choosing a ratio to find a missing side or angle

A right triangle has one angle of

90^\circ

. If you know one acute angle and one side, you can often find another side. If you know two sides, you can often find an acute angle. This lesson uses three primary trigonometric ratios: sine, cosine, and tangent. The main skill is matching the known and unknown sides to the correct ratio.

What you will learn

Identify the hypotenuse, opposite side, and adjacent side relative to a marked angle.
Choose and use sine, cosine, or tangent to solve a right-triangle problem.
Use inverse trigonometric functions to find an unknown angle.
Check whether a calculated answer is reasonable and include appropriate units.

1. Prerequisite bridge: name the sides

An acute angle is an angle greater than

0^\circ

and less than

90^\circ

. A right triangle has two acute angles. Choose one of them as the reference angle. The names opposite and adjacent depend on that choice.

The hypotenuse is the side opposite the

90^\circ

angle. It is the longest side. The opposite side is across from the reference angle. The adjacent side touches the reference angle and is not the hypotenuse.

For example, if the reference angle is marked

\theta

, locate the right angle first to identify the hypotenuse. Then use

\theta

to identify the opposite and adjacent sides. If you switch to the other acute angle, opposite and adjacent switch roles; the hypotenuse does not.

Find the hypotenuse by looking across from the right angle.
Find opposite and adjacent in relation to the chosen acute angle.
Side names describe positions, not fixed lengths.

2. The three primary trigonometric ratios

A trigonometric ratio compares the lengths of two sides of a right triangle. For a chosen acute angle

\theta

, sine compares opposite with hypotenuse, cosine compares adjacent with hypotenuse, and tangent compares opposite with adjacent.

A memory aid is SOH CAH TOA: sine uses Opposite over Hypotenuse, cosine uses Adjacent over Hypotenuse, and tangent uses Opposite over Adjacent. Use the ratio that contains both the side you know and the side you need.

For instance, if the known side is adjacent to the angle and the unknown side is opposite, tangent connects those two sides. Once you write the ratio, substitute the known values and solve the equation. Keep units with side lengths, and round only at the end when possible.

\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Sine: opposite divided by hypotenuse.
Cosine: adjacent divided by hypotenuse.
Tangent: opposite divided by adjacent.
Use the reference angle consistently when naming sides.

3. Solve for a side or an angle

To find a missing side, choose the ratio that includes the known angle, the known side, and the unknown side. Substitute and rearrange the equation. For example, from

\sin\theta=\frac{x}{h}

, multiply both sides by

h

to get

x=h\sin\theta

.

To find an angle when two sides are known, use the matching ratio and its inverse function. The inverse sine, written

\sin^{-1}

, returns an angle from a sine ratio. The inverse cosine and inverse tangent work in the same way. On a calculator, these may be labelled arcsin, arccos, and arctan.

Check that your calculator is in degree mode when the problem gives angles in degrees. An angle in a right triangle must be acute if it is one of the two angles other than the right angle. A side length must be positive.

\theta=\sin^{-1}\!\left(\frac{\text{opposite}}{\text{hypotenuse}}\right),\quad \theta=\cos^{-1}\!\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right),\quad \theta=\tan^{-1}\!\left(\frac{\text{opposite}}{\text{adjacent}}\right)

For a missing side, solve the ratio equation for the unknown length.
For a missing angle, use the inverse of the selected ratio.
Use degree mode for degree measures and round as the problem requests.

4. Apply the method and check the result

Right-triangle ratios can model a practical situation when the measurements form a right triangle. A ladder against a wall, for example, can form a right triangle with the wall and the ground. Label the angle and the sides before choosing a ratio.

A useful routine is: sketch or mark the right triangle, identify the reference angle, label known and unknown sides, select a ratio, solve, and check. The sketch does not need to be drawn to scale, but its labels must match the situation.

A calculated length should make sense compared with the other side lengths. The hypotenuse must be longer than either leg. A calculated acute angle should be between

0^\circ

and

90^\circ

. These checks can reveal a wrong ratio, a calculator-mode error, or a misplaced decimal.

Draw and label before calculating.
Match the ratio to the sides involved.
Check units, rounding, and whether the answer is possible.

Choose a ratio from the sides

Ratio	Side comparison	Use when the known and unknown sides are
Sine	Opposite over hypotenuse	Opposite and hypotenuse
Cosine	Adjacent over hypotenuse	Adjacent and hypotenuse
Tangent	Opposite over adjacent	Opposite and adjacent

Worked example

Find the height of a ramp

A straight ramp is

5.2

m long. It makes an angle of

28^\circ

with level ground. How high does the ramp rise? Round to the nearest tenth of a metre.

Identify the sides
The ramp is the hypotenuse because it is across from the right angle formed by the ground and vertical rise. The height is opposite the $28^\circ$ angle. Since the known and unknown sides are hypotenuse and opposite, use sine.
$\sin 28^\circ=\frac{h}{5.2}$
Solve for the height
Multiply both sides by $5.2$ to isolate $h$ . Evaluate with a calculator in degree mode.
$h=5.2\sin 28^\circ\approx 2.4405$
Round and interpret
Round the height to the nearest tenth. The result is positive and shorter than the ramp, as expected for a leg of the triangle.
$h\approx 2.4\text{ m}$

Answer: The ramp rises about

2.4

m.

Check: Because the height is opposite an acute angle, it must be shorter than the hypotenuse. The calculated value, about

2.44

m, is less than

5.2

m.

Common mistakes and how to avoid them

Calling a side opposite or adjacent without naming the reference angle.

Correction: Mark the acute angle you are using first. Opposite and adjacent are defined relative to that angle.

Using the wrong side pair in a trigonometric ratio.

Correction: Label the hypotenuse, opposite, and adjacent sides before selecting sine, cosine, or tangent.

Using a trigonometric ratio instead of its inverse to find an angle.

Correction: When the side ratio is known and the angle is unknown, use inverse sine, inverse cosine, or inverse tangent.

Rounding intermediate values too early or leaving off units.

Correction: Keep calculator values until the final step, then round as requested and state the correct unit for a length.

Lesson summary

Identify the hypotenuse from the right angle.
Name opposite and adjacent relative to the chosen acute angle.
Choose sine, cosine, or tangent based on the sides known and needed.
Use an inverse ratio to find an angle from two side lengths.
Check that the answer is reasonable and include units.

Check your understanding

Question 1

Relative to angle

\theta

, the opposite side is

6

cm and the hypotenuse is

10

cm. Which equation can be used to find

\theta

?

$\theta=\sin^{-1}(6/10)$
$\theta=\cos^{-1}(6/10)$
$\theta=\tan^{-1}(6/10)$
$\theta=\sin(6/10)$

Show answer and explanation

\theta=\sin^{-1}(6/10)

Sine compares opposite with hypotenuse, so the ratio is

6/10

. Use inverse sine because the angle is unknown.

Question 2

A right triangle has an angle of

35^\circ

and an adjacent side of

8

cm. Which expression gives the hypotenuse

h

?

$h=8\sin 35^\circ$
$h=8\cos 35^\circ$
$h=8/\cos 35^\circ$
$h=8\tan 35^\circ$

Show answer and explanation

h=8/\cos 35^\circ

Cosine is adjacent over hypotenuse, so

\cos 35^\circ=8/h

. Rearranging gives

h=8/\cos 35^\circ

.

Key terms

Acute angle: An angle greater than $0^\circ$ and less than $90^\circ$ .
Hypotenuse: The side opposite the right angle in a right triangle.
Opposite side: The side across from the chosen reference angle.
Adjacent side: The side beside the chosen reference angle that is not the hypotenuse.
Trigonometric ratio: A comparison of two side lengths in a right triangle, such as sine, cosine, or tangent.
Inverse trigonometric function: A calculator function used to find an angle when a suitable side ratio is known.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C1.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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