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

Side Ratios in Right Triangles: Sine, Cosine and Tangent

2.1About 70 minutes

Learning goals

I can name the hypotenuse, opposite side and adjacent side relative to an angle.
I can explain, using similar triangles, why a side ratio depends only on the angle.
I can write the sine, cosine and tangent ratios for an angle in a right triangle.
I can use a calculator in degree mode to find sin, cos and tan of an angle.

Words to know

hypotenuse
The side opposite the right angle. It is always the longest side of a right triangle.

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

adjacent side
The side that forms the angle you are working with and is not the hypotenuse.

sine (sin)
sin θ = opposite ÷ hypotenuse.

cosine (cos)
cos θ = adjacent ÷ hypotenuse.

tangent (tan)
tan θ = opposite ÷ adjacent.

θ (theta)
A Greek letter often used as the name of an angle.

trigonometry
The branch of math about relationships between the sides and angles of triangles. From Greek words for triangle and measure.

A ratio that belongs to an angle

In Lesson 2, three triangles with the same angles had the same side ratios, no matter how big they were. Now we focus on right triangles. Any two right triangles that share one acute angle θ are similar (AA: the right angle and θ). So for a given angle, each ratio of sides is always the same number. That number is so useful that it gets its own name. Welcome to trigonometry.

Naming the sides

In a right triangle, pick one of the acute angles and call it θ ("theta"). Then:

The hypotenuse is opposite the right angle. It never changes.
The opposite side is across from θ.
The adjacent side touches θ and is not the hypotenuse.

If you switch to the other acute angle, the opposite and adjacent sides swap.

The same right triangle twice. The hypotenuse stays the same, but "opposite" and "adjacent" depend on which angle you are looking from.

Investigation: a 35° angle

The diagram shows three right triangles that all have a 35° angle. The adjacent sides are 4, 6 and 8 cm. Measuring the opposite sides and the hypotenuses gives the table below.

Three similar right triangles with a shared 35° angle. However big the triangle, opposite ÷ adjacent ≈ 0.70.

Adjacent	Opposite	Hypotenuse	opp ÷ hyp	adj ÷ hyp	opp ÷ adj
4.0 cm	2.8 cm	4.9 cm	0.57	0.82	0.70
6.0 cm	4.2 cm	7.3 cm	0.58	0.82	0.70
8.0 cm	5.6 cm	9.8 cm	0.57	0.82	0.70

Measured to the nearest millimetre. Each ratio is (almost exactly) the same in all three triangles.

The three primary trigonometric ratios

For an acute angle θ in a right triangle:

sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent

(sin, cos and tan are short for sine, cosine and tangent.) Many students remember them with the word SOH CAH TOA.

SOH CAH TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

Using a calculator

Your calculator knows these ratios for every angle. First make sure it is in degree mode (look for DEG on the screen). Test it: sin 30° should give 0.5. If you get −0.988..., it is in radian mode; change the mode.

Try sin 35°, cos 35° and tan 35°. You should get 0.5736, 0.8192 and 0.7002 (to four decimal places). These match the measured ratios in the table. Usually we round trig ratios to four decimal places.

Worked example

Example 1: Writing all six ratios

In △ABC, ∠C = 90°, BC = 3, AC = 4 and AB = 5. Write sin, cos and tan for ∠A and for ∠B.

Hypotenuse: AB = 5 (opposite the right angle).
For ∠A: opposite = BC = 3, adjacent = AC = 4. sin A = 3/5 = 0.6, cos A = 4/5 = 0.8, tan A = 3/4 = 0.75.
For ∠B: opposite = AC = 4, adjacent = BC = 3. sin B = 4/5 = 0.8, cos B = 3/5 = 0.6, tan B = 4/3 ≈ 1.3333.
Notice: sin A = cos B and cos A = sin B. The side opposite one acute angle is adjacent to the other.

sin A = 0.6, cos A = 0.8, tan A = 0.75; sin B = 0.8, cos B = 0.6, tan B ≈ 1.3333.

Worked example

Example 2: Find a side first

A right triangle has legs 5 cm and 12 cm. θ is the angle opposite the 5 cm side. Find sin θ, cos θ and tan θ.

Pythagorean theorem: hypotenuse² = 5² + 12² = 25 + 144 = 169, so the hypotenuse is 13 cm.
Opposite θ: 5. Adjacent to θ: 12. Hypotenuse: 13.
sin θ = 5/13 ≈ 0.3846, cos θ = 12/13 ≈ 0.9231, tan θ = 5/12 ≈ 0.4167.

sin θ ≈ 0.3846, cos θ ≈ 0.9231, tan θ ≈ 0.4167.

Useful facts

The hypotenuse is the longest side, so for an acute angle sin θ and cos θ are always between 0 and 1.
tan θ can be any positive number. tan 45° = 1 (the two legs are equal), and tan θ > 1 when θ > 45°.
If two acute angles add to 90° (they are complementary), the sine of one equals the cosine of the other: for example, sin 40° = cos 50° ≈ 0.6428.
Trig ratios have no units: they are a length divided by a length.

Just for fun

Some students remember SOH CAH TOA with the sentence "Some Old Hippo Caught Another Hippo Taking Oats Away." Silly sentences are easier to remember, so make up your own!

Hands-on activity

Discover tan 35° with a ruler

You need: a protractor, a ruler, a sharp pencil and paper.

Draw a horizontal line about 15 cm long. At its left end, use the protractor to draw a 35° angle with a long arm.
At 5 cm, 10 cm and 14 cm from the vertex along the horizontal line, draw vertical lines (use the protractor's 90° mark) up to the slanted arm.
For each of the three right triangles, measure the opposite side (the vertical line) and the hypotenuse (along the slanted arm).
Calculate opposite ÷ adjacent and opposite ÷ hypotenuse for each.
Compare with tan 35° and sin 35° on a calculator.

What to look for: opposite ÷ adjacent should be close to 0.70 each time, and opposite ÷ hypotenuse close to 0.57. Then repeat with a 60° angle and predict: will opposite ÷ adjacent be bigger or smaller? (Bigger: tan 60° ≈ 1.73.)

Your turn

In right △ABC with ∠C = 90°, name the hypotenuse and the sides opposite and adjacent to ∠A.

A right triangle has sides 8, 15 and 17. Angle θ is opposite the side of length 8. Write sin θ, cos θ and tan θ as fractions.

In the same triangle, α is the other acute angle. Write sin α, cos α and tan α.

Use a calculator to find sin 52°, cos 52° and tan 52° to four decimal places.

A right triangle has legs 7 and 24. Find the hypotenuse and the sine of the smallest angle.

Explain why the sine of an acute angle can never be greater than 1.

Which acute angle has tan θ = 1? Explain.

More practice: warm-up

Write what each part of SOH CAH TOA stands for.

Which side of a right triangle is always the longest?

Opposite = 6 and adjacent = 8. Find tan θ.

What does a calculator in degree mode give for cos 60°?

Which ratio uses the opposite side and the hypotenuse?

More practice: on your own

A right triangle has hypotenuse 10 and the side opposite θ is 6. Find the adjacent side and all three ratios for θ.

Ramp A rises 1 m over a horizontal distance of 4 m. Ramp B rises 2.5 m over 10 m. Do they make the same angle with the ground? Explain.

Use a calculator to check that sin 40° = cos 50°. Explain why using a right triangle.

Find (sin 30°)² + (cos 30°)². Explain the answer using the Pythagorean theorem.

In △PQR, ∠Q = 90°, PQ = 9 and QR = 12. Find PR, tan R and sin P.

Challenge

Show that tan θ = sin θ ÷ cos θ for any acute angle in a right triangle.

An isosceles right triangle has legs of length 1. Find the exact hypotenuse and the exact value of sin 45°. Check with a calculator.

Cut an equilateral triangle with sides 2 in half to make a right triangle with angles 30°, 60° and 90°. Find its side lengths and the exact values of sin 30°, cos 30° and tan 60°.

Check your understanding

Quiz

In a right triangle, sin θ equals:

A right triangle has sides 3, 4 and 5. What is the tangent of the angle opposite the side of length 3?

Why does every right triangle with a 35° angle have the same value of tan 35°?

What is tan 45°?

For an acute angle θ, which ratio can be greater than 1?

The adjacent side is 12 and the hypotenuse is 13. What is cos θ?

For parents and teachers

Big idea: Expectation 2.1: students investigate why the ratio of two sides is the same in similar right triangles, and define sine, cosine and tangent.

How to help: Check the calculator first: it must be in degree mode (sin 30° = 0.5). Point at a triangle and ask "Which side is opposite this angle? Which is adjacent?" Labelling the sides correctly is the step most mistakes come from.

Common difficulties:

Calling the hypotenuse "adjacent" because it touches the angle. The hypotenuse is always named first and is never opposite or adjacent.
Forgetting that opposite and adjacent swap when you switch angles.
Calculator in radian mode (answers look strange or negative).
Thinking sin, cos and tan are numbers you multiply by ("sin × 30"). They are functions of the angle: sin 30° is one number.