Grade by Grade
D. Trigonometric Functions · Trigonometric Ratios, Identities and Laws

Exact Values: Special Triangles and Special Angles

D1.1About 65 minutes
Learning goals
  • I can build the 45-45-90 and 30-60-90 special triangles from a square and an equilateral triangle.
  • I can state the exact values of sine, cosine and tangent for 0°, 30°, 45°, 60° and 90°.
  • I can evaluate expressions with special-angle ratios exactly, using radicals and rationalized denominators.
  • I can use exact values to find exact side lengths in right triangles.

Words to know

exact value
A value written without rounding, using integers, fractions and radicals, such as √3/2. A decimal like 0.866 is only an approximation.
special angles
The angles 0°, 30°, 45°, 60° and 90°, whose trigonometric ratios have simple exact values.
special triangles
The 45-45-90 triangle (sides 1, 1, √2) and the 30-60-90 triangle (sides 1, √3, 2).
rationalize the denominator
Rewrite a fraction so that no radical is left in the denominator, for example 1/√2 = √2/2.
sin²θ
Short for (sin θ)², the square of the sine of θ. It is not sin(θ²).

Why exact values?

Type sin 60° into your calculator and you get 0.8660254038. That is a rounded decimal. The exact value is √3/2, and it never needs rounding. Exact values matter because rounding early makes errors grow, and because many problems in this course (and in Grade 12 Advanced Functions, MHF4U, and Calculus and Vectors, MCV4U) expect answers like 5√3 m instead of 8.66 m.

Remember from Grade 10: in a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent (SOH CAH TOA). In this lesson you will find the ratios for a few special angles exactly, using two triangles you can build from shapes you already know.

The 45-45-90 triangle

Start with a square with side 1. Cut it along a diagonal. Each half is an isosceles right triangle with two 45° angles.

By the Pythagorean theorem, the diagonal is √(1² + 1²) = √2. So the sides are 1, 1, √2.

For 45°, the opposite side and the adjacent side are both 1, and the hypotenuse is √2:

  • sin 45° = 1/√2
  • cos 45° = 1/√2
  • tan 45° = 1/1 = 1

We usually rationalize the denominator: multiply the top and bottom by √2 to get 1/√2 = √2/2. (You practised simplifying radicals in the unit Functions and Equivalent Expressions.)

11√245°a square with side 145°45°1√21sin 45° = 1/√2 = √2/2cos 45° = 1/√2 = √2/2tan 45° = 1/1 = 1
A square with side 1 cut along its diagonal makes two 45-45-90 triangles. Both legs are 1 and the hypotenuse is √2.

The 30-60-90 triangle

Start with an equilateral triangle with side 2. All its angles are 60°. Draw the altitude from the top vertex. It cuts the base into two halves of length 1 and splits the top angle into two 30° angles.

Each half is a right triangle with angles 30°, 60° and 90°. Its hypotenuse is 2, its short leg is 1, and the long leg is √(2² − 1²) = √3. So the sides are 1, √3, 2.

  • The short side (1) is opposite the 30° angle.
  • The middle side (√3) is opposite the 60° angle.
  • The hypotenuse (2) is opposite the right angle.
221160°60°equilateral, side 260°30°√321sin 30° = 1/2cos 30° = √3/2tan 30° = 1/√3 = √3/3sin 60° = √3/2cos 60° = 1/2tan 60° = √3/1 = √3
Half of an equilateral triangle with side 2 is a 30-60-90 triangle with sides 1, √3 and 2. Read each ratio straight from the triangle.

What about 0° and 90°?

Imagine a right triangle whose hypotenuse is always 1. Then sin θ is just the length of the opposite side, and cos θ is the length of the adjacent side.

  • As θ shrinks toward 0°, the opposite side shrinks to 0 and the adjacent side stretches to 1. So sin 0° = 0, cos 0° = 1 and tan 0° = 0/1 = 0.
  • As θ grows toward 90°, the opposite side grows to 1 and the adjacent side shrinks to 0. So sin 90° = 1 and cos 90° = 0. Then tan 90° = 1/0, and you cannot divide by 0, so tan 90° is undefined.

In the next lesson, angles in standard position on a coordinate grid will make these values even clearer.

θsin θ1cos θθ = 70°θsin θ1cos θθ = 30°sin θ1cos θθ = 5°
Each hypotenuse is 1. As θ gets smaller, the vertical side (sin θ) shrinks toward 0 and the horizontal side (cos θ) grows toward 1.
θ0°30°45°60°90°
sin θ01/2√2/2√3/21
cos θ1√3/2√2/21/20
tan θ0√3/3 (= 1/√3)1√3undefined
Exact values for the special angles. Learn this table well; the rest of the unit uses it constantly.

A pattern to help you remember

Write the sines as √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. These simplify to 0, 1/2, √2/2, √3/2, 1. The cosines are the same list backwards. And tan θ = sin θ ÷ cos θ in every column (for example, tan 60° = (√3/2) ÷ (1/2) = √3).

Worked example

Example 1: Evaluating an expression exactly

Find the exact value of sin 30° cos 60° + sin 60° cos 30°.

  1. Replace each ratio with its exact value: (1/2)(1/2) + (√3/2)(√3/2).
  2. Multiply: 1/4 + 3/4. (Remember √3 × √3 = 3.)
  3. Add: 4/4 = 1.

1 (which happens to equal sin 90°).

Worked example

Example 2: Exact side lengths

A wheelchair ramp is 6 m long and makes an angle of 30° with the level ground. Find the exact height it rises and the exact horizontal distance it covers.

  1. Sketch a right triangle: hypotenuse 6 m, angle 30° at the bottom.
  2. Height is opposite 30°: height = 6 sin 30° = 6 × 1/2 = 3.
  3. Horizontal distance is adjacent to 30°: 6 cos 30° = 6 × √3/2 = 3√3.
  4. Check with decimals: 3√3 ≈ 5.20, and 3² + 5.20² ≈ 9 + 27 = 36 = 6². ✓

Height 3 m; horizontal distance 3√3 m (about 5.2 m). (A real ramp would be far less steep; Ontario building rules require much gentler slopes. This one just makes the arithmetic clear.)

Worked example

Example 3: Finding a hypotenuse exactly

In △PQR, ∠Q = 90°, ∠P = 60° and PQ = 7 cm. Find PR and QR exactly.

  1. PQ is adjacent to ∠P, and PR is the hypotenuse. cos 60° = PQ/PR, so 1/2 = 7/PR, and PR = 14.
  2. QR is opposite ∠P. tan 60° = QR/PQ, so QR = 7 tan 60° = 7√3.
  3. Check with the 1 : √3 : 2 pattern: the side opposite 30° (∠R) is PQ = 7, so the sides are 7, 7√3, 14. ✓

PR = 14 cm, QR = 7√3 cm

Worked example

Example 4: Squares of ratios

Evaluate sin²45° + cos²45° and sin²60° + cos²60° exactly.

  1. sin²45° = (√2/2)² = 2/4 = 1/2, and cos²45° is also 1/2. Sum: 1.
  2. sin²60° = (√3/2)² = 3/4 and cos²60° = (1/2)² = 1/4. Sum: 1.

Both equal 1. This is no accident. In lesson 6 (Trigonometric Identities) you will prove that sin²θ + cos²θ = 1 for every angle.

Just for fun

Your calculator says sin 60° = 0.8660254038. The exact value √3/2 is shorter to write, never needs rounding, and never runs out of battery.

Hands-on activity

Fold the special triangles

You need: a sheet of printer paper, a ruler, a pencil, a protractor, scissors (use them carefully) and a calculator.

  1. Make a square. Fold one corner of the sheet down to the opposite edge so the top edge lines up with the side edge. Cut off the extra strip. Unfold: you have a square with a diagonal crease.
  2. Measure the side s and the diagonal d to the nearest millimetre. Calculate d ÷ s. Compare with √2 ≈ 1.414.
  3. Make an equilateral triangle. Draw a 12 cm segment. Use the protractor to draw 60° angles at both ends, and extend the arms until they meet. Check that all three sides are 12 cm.
  4. Fold the triangle in half so the base ends meet. The crease is the altitude. Measure it and divide by 6 (half the base). Compare with √3 ≈ 1.732.
  5. Use your measurements to estimate sin 60° (altitude ÷ 12) and compare with √3/2 ≈ 0.866.

What to look for: d ÷ s should be between about 1.40 and 1.43. The altitude should be close to 10.4 cm, since 6√3 ≈ 10.39. Small differences come from measuring and folding.

Your turn

1

Find the exact value of sin 45° cos 45°.

2

Find the exact value of tan 30° tan 60°.

3

Find the exact value of 2 sin 30° + cos 0°.

4

Find the exact value of sin²60° + cos²60°.

5

A right isosceles triangle has hypotenuse 8 cm. Find the exact length of each leg.

6

Rationalize the denominator: 1/√3.

7

Find the exact value of tan 60° − 2 sin 60°.

More practice: warm-up

1

sin 30° = ?

2

cos 45° = ?

3

tan 60° = ?

4

sin 90° = ?

5

cos 90° = ?

More practice: on your own

1

Find the exact value of cos²30° − sin²30°.

2

Find the exact value of sin 60° ÷ cos 60°. Which ratio does it equal?

3

A 4 m ladder leans against a wall and makes a 60° angle with the ground. Find, exactly, how high up the wall it reaches and how far its foot is from the wall.

4

In △PQR, ∠Q = 90°, ∠P = 30° and QR = 5. Find PR and PQ exactly.

5

Is sin 60° equal to 2 sin 30°? Explain.

Challenge

1

Show that (tan 45° + tan 60°) ÷ (1 − tan 45° tan 60°) = −2 − √3.

2

An equilateral triangle has side s. Find its height and its area exactly, in terms of s.

3

A regular hexagon has sides of 6 cm. Find its exact area. (Hint: it is made of six equilateral triangles.)

Check your understanding

Quiz

1.

What is the exact value of cos 30°?

2.

What is tan 45°?

3.

Which value equals tan 30°?

4.

What is sin²45° + cos²45°?

5.

In a 30-60-90 triangle the shortest side is 4 cm. How long is the hypotenuse?

6.

What is tan 90°?

For parents and teachers

Big idea (D1.1): the exact values of sine, cosine and tangent for 0°, 30°, 45°, 60° and 90° come from two triangles: half a square (sides 1, 1, √2) and half an equilateral triangle (sides 1, √3, 2). Your teen should be able to draw these triangles from memory rather than memorize a table blindly.

How to help: Ask your teen to sketch both triangles on a scrap of paper and read off a ratio you name ("What is cos 60°?"). Then check with a calculator in degree mode: cos 60° should give 0.5.

Common difficulties:

  • Mixing up which side is opposite 30° and which is opposite 60° (the shorter side is always opposite the smaller angle).
  • Arithmetic with radicals: √3 × √3 = 3, and (√2/2)² = 2/4 = 1/2.
  • Writing sin²θ as sin(θ²). sin²θ means (sin θ)².
  • Forgetting that tan 90° is undefined, not 0.