T1 Use sine, cosine, and tangent in right triangles

Ontario Grade 10 Mathematics

Trigonometry

Choose a right-triangle ratio to find a missing side or angle

A right triangle is a triangle with one

90^\circ

angle. You may already know how to find the perimeter or area of a triangle. Sine, cosine, and tangent are three ratios that connect an acute angle to the side lengths of a right triangle. An acute angle is greater than

0^\circ

and less than

90^\circ

. These ratios can help you find a missing side or angle when you know other measurements.

What you will learn

Identify the opposite, adjacent, and hypotenuse sides in relation to a chosen angle.
Choose sine, cosine, or tangent to connect known and unknown sides.
Use a calculator to find a missing side or angle in a right triangle.
Check that an answer is reasonable and includes the correct units.

1. Review the triangle and name its sides

Begin by finding the angle named in the question. The names of two sides depend on which acute angle you use. If you switch to the other acute angle, the opposite and adjacent sides switch roles.

The hypotenuse is the side across from the right angle. It is also the longest side. The opposite side is across from the chosen acute angle. The adjacent side touches the chosen angle but is not the hypotenuse.

Picture a ramp that rises from the ground. The ramp, the ground, and the vertical rise form a right triangle. If you choose the angle where the ramp meets the ground, the ramp is the hypotenuse, the vertical rise is opposite, and the ground distance is adjacent. This real situation gives you a way to name the sides before you calculate.

Each trigonometric ratio compares two side lengths. Sine compares opposite with hypotenuse. Cosine compares adjacent with hypotenuse. Tangent compares opposite with adjacent. The memory phrase SOH CAH TOA can help you recall these pairings. Read it as three short reminders: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

\sin\theta=\frac{O}{H},\quad \cos\theta=\frac{A}{H},\quad \tan\theta=\frac{O}{A}

Mark the chosen acute angle before naming sides.
The hypotenuse is always across from the right angle.
Opposite and adjacent are named in relation to the chosen angle.

2. Choose a ratio to find a side

The symbol

\theta

(theta) can stand for an angle. In the ratios,

O

means opposite,

A

means adjacent, and

H

means hypotenuse. A ratio is a comparison made by dividing one quantity by another.

To choose a ratio, identify the angle, the known side, and the unknown side. Then select the ratio that includes both sides. For example, use sine if the opposite side and hypotenuse are involved. Use cosine if the adjacent side and hypotenuse are involved. Use tangent if the opposite and adjacent sides are involved.

Write the ratio before using your calculator. Substitute the angle and the known side, then rearrange the equation to make the unknown side stand alone. For example, if sine equals the opposite side divided by the hypotenuse, multiply or divide both sides as needed to isolate the unknown. Use a calculator in degree mode when the question gives an angle in degrees.

Keep the units with your side measurements. If the question asks for a length to the nearest tenth, keep extra calculator digits during the work and round the final answer only. This avoids small rounding differences from building up in later calculations.

\sin\theta=\frac{O}{H},\quad \cos\theta=\frac{A}{H},\quad \tan\theta=\frac{O}{A}

Use sine for opposite and hypotenuse.
Use cosine for adjacent and hypotenuse.
Use tangent for opposite and adjacent.
Choose a ratio by matching the side pair, not by guessing from a picture.

3. Use side lengths to find an angle

Sometimes the side lengths are known and the angle is missing. First choose the ratio that contains the two known sides. Then use the matching inverse ratio function on a calculator. An inverse ratio function works backward: it finds an angle from a side ratio.

For example, if the opposite and adjacent sides are known, divide the opposite length by the adjacent length. Then use inverse tangent to find the angle. On a calculator, this function may appear as

\tan^{-1}

or as an inverse option reached with a key such as SHIFT. The symbols

\sin^{-1}

and

\cos^{-1}

work in the same way for their matching side pairs.

The superscript

-1

in these symbols means inverse function here. It does not mean that you should raise a side ratio to the power of negative one. Check that the calculator is in degree mode. An answer for an acute angle in a right triangle should be between

0^\circ

and

90^\circ

.

Check whether your result makes sense. The hypotenuse must be longer than either of the other sides. Sine and cosine of an acute angle are positive and less than

1

, because each is a shorter side divided by the hypotenuse. A calculator result outside a reasonable range can signal a side-label, ratio, or calculator-mode error.

\theta=\sin^{-1}\left(\frac{O}{H}\right),\quad \theta=\cos^{-1}\left(\frac{A}{H}\right),\quad \theta=\tan^{-1}\left(\frac{O}{A}\right)

Use an inverse ratio when the unknown is an angle and two sides are known.
Match inverse sine, inverse cosine, or inverse tangent to the known side pair.
Check the angle mode, side lengths, units, and rounding.

4. Guided example and independent practice

In a word problem, sketching the right triangle can make the side relationships clearer. Mark the right angle and the angle named in the question. Label the sides in relation to that angle. Then choose a ratio and write an equation before calculating.

In the guided example, the ramp’s angle with the ground identifies the angle to use. The rise is opposite, the ramp is the hypotenuse, and the ground distance is adjacent. The first calculation uses sine because it connects the known rise to the unknown ramp length. The second uses cosine because it connects the ground distance to the ramp length.

Now practise the same decision-making steps. For each question, sketch and label the triangle, choose the ratio, and show the equation before using a calculator. (1) A right triangle has an angle of

42^\circ

and a hypotenuse of

9

cm. Find the opposite side to the nearest tenth. (2) A right triangle has opposite and adjacent sides of

7

cm and

10

cm. Find the angle to the nearest degree. (3) A right triangle has an angle of

28^\circ

and an adjacent side of

6

m. Find the hypotenuse to the nearest tenth.

Sketch and label before calculating.
Write the ratio that matches the known and unknown sides.
Use an inverse ratio when you need an angle.

Choosing a trigonometric ratio

Ratio	Side pair	Use it when
Sine	Opposite and hypotenuse	The known and unknown sides are this pair
Cosine	Adjacent and hypotenuse	The known and unknown sides are this pair
Tangent	Opposite and adjacent	The known and unknown sides are this pair

Worked example

Find the ramp length and ground distance

A ramp makes an angle of

31^\circ

with level ground. The vertical rise is

2.4

m. Find the ramp length and the horizontal ground distance, to the nearest tenth of a metre.

Name the sides
Use the angle where the ramp meets the ground. The vertical rise is opposite this angle, the ramp is the hypotenuse, and the ground distance is adjacent. The ramp length is the unknown in the first calculation.
Find the ramp length
The rise is opposite and the ramp is the hypotenuse, so use sine. Rearrange the ratio to isolate the hypotenuse. Evaluate with the calculator in degree mode.
$\sin 31^\circ=\frac{2.4}{H},\quad H=\frac{2.4}{\sin 31^\circ}\approx 4.66\ \mathrm{m}$
Find the ground distance
The ground distance is adjacent and the ramp is the hypotenuse, so use cosine. Use the unrounded ramp length in the calculation, then round the final distance.
$\cos 31^\circ=\frac{A}{H},\quad A\approx 4.66\cos 31^\circ\approx 4.0\ \mathrm{m}$
Check the result
The ramp is longer than either leg of the triangle, as expected for a hypotenuse. Both the rise and the ground distance are shorter than the ramp. The ground distance is longer than the rise, which fits the relatively shallow angle.

Answer: The ramp is approximately

4.7

m long, and the horizontal ground distance is approximately

4.0

m.

Check: The side lengths are about

2.4

m,

4.0

m, and

4.7

m. The longest is the ramp, which is the hypotenuse.

Common mistakes and how to avoid them

Calling a side adjacent just because it looks beside the angle.

Correction: Find the hypotenuse first. It is across from the right angle. Then name opposite and adjacent in relation to the chosen acute angle.

Choosing a ratio because its name seems familiar without checking the sides.

Correction: Name the known and unknown sides first. Select the ratio that includes both.

Using a regular ratio instead of an inverse ratio to find an angle.

Correction: When the unknown is an angle, use the matching inverse function. For opposite and adjacent, use inverse tangent.

Leaving the calculator in radian mode or rounding too early.

Correction: Set the calculator to degrees when the angle is written with a degree sign. Keep extra digits until the final answer.

Lesson summary

Mark the right angle and chosen acute angle before naming sides.
Use SOH CAH TOA to match a side pair with sine, cosine, or tangent.
Rearrange a ratio to find a missing side; use an inverse ratio to find a missing angle.
Check the answer’s size, units, and rounding.

Check your understanding

Question 1

A right triangle has an angle of

35^\circ

and a hypotenuse of

8

cm. Which equation finds the side opposite the angle,

x

?

$\sin 35^\circ=\frac{x}{8}$
$\cos 35^\circ=\frac{x}{8}$
$\tan 35^\circ=\frac{x}{8}$
$\sin 35^\circ=\frac{8}{x}$

Show answer and explanation

\sin 35^\circ=\frac{x}{8}

Sine compares opposite with hypotenuse. Here,

x

is opposite and

8

cm is the hypotenuse.

Question 2

A right triangle has opposite and adjacent sides of

5

cm and

12

cm relative to an angle

\theta

. Which expression finds

\theta

?

$\sin^{-1}\left(\frac{5}{12}\right)$
$\cos^{-1}\left(\frac{5}{12}\right)$
$\tan^{-1}\left(\frac{5}{12}\right)$
$\tan^{-1}\left(\frac{12}{5}\right)$

Show answer and explanation

\tan^{-1}\left(\frac{5}{12}\right)

Tangent compares opposite with adjacent, so the ratio is

\frac{5}{12}

. Inverse tangent gives the angle.

Question 3

A right triangle has an angle of

50^\circ

and an adjacent side of

7

m. Which ratio would find the hypotenuse

h

?

$\sin 50^\circ=\frac{7}{h}$
$\cos 50^\circ=\frac{7}{h}$
$\tan 50^\circ=\frac{7}{h}$
$\cos 50^\circ=\frac{h}{7}$

Show answer and explanation

\cos 50^\circ=\frac{7}{h}

Cosine compares adjacent with hypotenuse. The known adjacent side is

7

m, so

\cos 50^\circ=\frac{7}{h}

.

Key terms

Right triangle: A triangle with one angle measuring $90^\circ$ .
Hypotenuse: The side across from the right angle in a right triangle.
Opposite side: The side across from the chosen acute angle.
Adjacent side: The side beside the chosen acute 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 ratio function: A calculator operation, such as $\sin^{-1}$ , that finds an angle from a side ratio.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic T1. It is a study resource, not an official curriculum publication.

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T1 · Use sine, cosine, and tangent in right triangles

Ontario Grade 10 Mathematics

Choose a right-triangle ratio to find a missing side or angle

What you will learn

1. Review the triangle and name its sides

2. Choose a ratio to find a side

3. Use side lengths to find an angle

4. Guided example and independent practice

Choosing a trigonometric ratio

Find the ramp length and ground distance

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MPM2D

About this lesson