Define sine, cosine, and tangent from similar right triangles

Ontario Grade 10 Mathematics

Measurement and Trigonometry

How side-length ratios describe an angle

A right triangle has one angle of

90^\circ

. Its side lengths can change while its angles and shape stay the same. In this lesson, you will use that idea to define three ratios: sine, cosine, and tangent. A ratio compares two quantities by division. These ratios describe a chosen angle in a right triangle; they are not names for the triangle’s side lengths.

What you will learn

Identify the hypotenuse, opposite side, and adjacent side for a chosen angle in a right triangle.
Explain why matching side ratios stay the same in similar right triangles.
Define sine, cosine, and tangent as ratios of side lengths.
Choose the correct ratio name for a stated angle and pair of sides.

1. Grade 9 bridge: sides and similar triangles

First, review two ideas. The longest side of a right triangle is across from the

90^\circ

angle. It is called the hypotenuse. The other two sides meet to form the right angle.

Similar triangles have the same shape, though they may be different sizes. Their matching angles are equal, and their matching side lengths have the same scale factor. For example, if every side in one triangle is multiplied by

2

, the new triangle is similar to the first.

Now choose one of the two angles that is not the right angle. The names of the other sides depend on this choice. The opposite side is across from the chosen angle. The adjacent side touches the chosen angle and is not the hypotenuse. The hypotenuse keeps its name no matter which acute angle you choose.

An acute angle is greater than

0^\circ

and less than

90^\circ

. The two non-right angles in a right triangle are acute. In the rest of the lesson, the chosen angle will be an acute angle labelled

\theta

(the Greek letter theta).

The hypotenuse is across from the right angle and is the longest side.
Opposite and adjacent are named in relation to the chosen angle.
Similar triangles have matching angles and matching side lengths in the same proportions.

2. From a concrete comparison to three ratios

Imagine two right triangles with the same acute angle,

\theta

. In the smaller triangle, the opposite, adjacent, and hypotenuse side lengths are

3

,

4

, and

5

. In a larger, similar triangle, those lengths are

6

,

8

, and

10

. Each matching side in the larger triangle is twice as long.

Compare opposite to hypotenuse in each triangle. The smaller triangle gives

3/5

, and the larger gives

6/10

. These are equal because both side lengths in the second comparison were multiplied by

2

. The same idea works for the other matching side pairs: adjacent to hypotenuse gives

4/5

and

8/10

, while opposite to adjacent gives

3/4

and

6/8

.

This is why the ratio for a particular angle does not depend on the size of the triangle. Any right triangles with that same acute angle are similar, so their corresponding sides grow or shrink by the same factor. The quotient, or result of division, for each matching pair stays unchanged.

The three ratios have names. Sine, written

\sin \theta

, is opposite divided by hypotenuse. Cosine, written

\cos \theta

, is adjacent divided by hypotenuse. Tangent, written

\tan \theta

, is opposite divided by adjacent. These definitions apply to the chosen angle

\theta

.

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

Sine compares the opposite side with the hypotenuse.
Cosine compares the adjacent side with the hypotenuse.
Tangent compares the opposite side with the adjacent side.
A ratio stays the same for similar right triangles with the same chosen angle.

3. Guided example: name each ratio for an angle

Suppose a right triangle has side lengths

5

,

12

, and

13

. The side of length

13

is the hypotenuse because it is the longest side. Choose the acute angle across from the side of length

5

and call it

\theta

.

Relative to this angle, the side of length

5

is opposite. The side of length

12

touches the angle and is not the hypotenuse, so it is adjacent. Once the side roles are clear, use the definitions to write each ratio. No angle measure is needed to define these ratios.

Choose the angle before labelling opposite and adjacent.
The hypotenuse is identified from the right angle, not from the chosen acute angle.

4. Independent practice and a careful check

Practise the definitions without calculating an angle or finding a missing side. For a new right triangle, mark the chosen acute angle, identify the hypotenuse, and then name the opposite and adjacent sides. Only after that should you select sine, cosine, or tangent.

If the chosen angle changes to the other acute angle, the hypotenuse stays the same but the opposite and adjacent sides switch roles. As a result, the sine and cosine descriptions use the other leg, and tangent compares the two legs in the opposite order. This is not a change to the definitions; it is a change in which angle you are describing.

When triangles are drawn in different sizes, do not assume the ratios are different just because the side lengths look larger. Compare matching sides. Similarity means the same scale factor applies to every corresponding side, so dividing one matching length by another gives the same ratio.

For independent practice, sketch any right triangle and label one acute angle

\theta

. Write which side is opposite, adjacent, and the hypotenuse. Then write the three ratios using those labels. Check that the denominator in sine and cosine is the hypotenuse, while tangent uses the adjacent side as its denominator.

A reliable order is: choose the angle, identify the hypotenuse, label opposite and adjacent, then write ratios.
Use side roles relative to the specified angle, not just the apparent position of a side on the page.
The definitions compare lengths; they do not require a particular triangle size.

Side ratios in two similar right triangles

Side comparison	Smaller triangle	Larger triangle
Opposite to hypotenuse	$3/5$	$6/10$
Adjacent to hypotenuse	$4/5$	$8/10$
Opposite to adjacent	$3/4$	$6/8$

Worked example

Find the three ratios from side roles

A right triangle has side lengths

5

,

12

, and

13

. For the acute angle

\theta

across from the side of length

5

, write

\sin\theta

,

\cos\theta

, and

\tan\theta

.

Identify the hypotenuse
The hypotenuse is across from the right angle and is the longest side. Here it has length $13$ .
$13$
Name the other sides
The chosen angle is across from the side of length $5$ , so that side is opposite. The remaining side of length $12$ touches the angle and is not the hypotenuse, so it is adjacent.
$\text{opposite}=5,\quad \text{adjacent}=12$
Apply the definitions
Sine uses opposite over hypotenuse, cosine uses adjacent over hypotenuse, and tangent uses opposite over adjacent.
$\sin\theta=\frac{5}{13},\quad \cos\theta=\frac{12}{13},\quad \tan\theta=\frac{5}{12}$

Answer:

\sin\theta=\frac{5}{13},\quad \cos\theta=\frac{12}{13},\quad \tan\theta=\frac{5}{12}

Check: The chosen angle is across from the side of length

5

, so that is the opposite side. The hypotenuse is

13

, leaving

12

as the adjacent side. Each ratio uses the correct pair of sides.

Common mistakes and how to avoid them

Calling a side opposite because it appears across the page from the angle.

Correction: Opposite means directly across from the chosen angle. Turn or rotate the triangle on the page and the side roles do not change.

Calling the hypotenuse adjacent because it touches the chosen angle.

Correction: The adjacent side is the side that touches the chosen angle but is not the hypotenuse. The hypotenuse always lies across from the right angle.

Using the same opposite and adjacent sides after changing the chosen angle.

Correction: Recheck the sides relative to the new angle. The hypotenuse stays the same, but opposite and adjacent switch.

Thinking a larger triangle must have different sine, cosine, and tangent ratios.

Correction: If the triangles are similar and have the same chosen angle, corresponding side lengths share a scale factor. That factor cancels in each side-length ratio.

Lesson summary

For a chosen acute angle, opposite and adjacent name the two non-hypotenuse sides relative to that angle.
Sine is opposite divided by hypotenuse; cosine is adjacent divided by hypotenuse; tangent is opposite divided by adjacent.
Similar right triangles with the same acute angle have the same corresponding side ratios, even when their sizes differ.

Check your understanding

Question 1

For angle

\theta

, the opposite side is

7

and the hypotenuse is

25

. Which expression defines

\sin\theta

?

$\frac{7}{25}$
$\frac{25}{7}$
$\frac{7}{\text{adjacent}}$
$\frac{\text{adjacent}}{25}$

Show answer and explanation

\frac{7}{25}

Sine is the opposite side divided by the hypotenuse, so the ratio is

7/25

.

Question 2

For angle

\phi

, the adjacent side is

8

and the hypotenuse is

17

. Which expression defines

\cos\phi

?

$\frac{17}{8}$
$\frac{8}{17}$
$\frac{\text{opposite}}{8}$
$\frac{8}{\text{opposite}}$

Show answer and explanation

\frac{8}{17}

Cosine is adjacent divided by hypotenuse. The given side lengths make that ratio

8/17

.

Question 3

For angle

\alpha

, the opposite side is

9

and the adjacent side is

12

. Which expression defines

\tan\alpha

?

$\frac{12}{9}$
$\frac{9}{\text{hypotenuse}}$
$\frac{9}{12}$
$\frac{12}{\text{hypotenuse}}$

Show answer and explanation

\frac{9}{12}

Tangent is opposite divided by adjacent, so the ratio is

9/12

.

Key terms

Right triangle: A triangle with one angle measuring $90^\circ$ .
Hypotenuse: The longest side of a right triangle, across from the right angle.
Opposite side: The side across from the chosen angle.
Adjacent side: The side that touches the chosen angle and is not the hypotenuse.
Similar triangles: Triangles with the same shape: their matching angles are equal and their matching side lengths have the same scale factor.
Ratio: A comparison of two quantities, often made by dividing one by the other.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic M4. It is a study resource, not an official curriculum publication.

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