Special Triangles

Learn 30-60-90 and 45-45-90 triangles and their ratios. Math 20-1 Alberta curriculum.

Lesson 4.4 of Trigonometry in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Two Special Triangles. Learn the 45-45-90 and 30-60-90 triangles and derive the exact trig values for 30°, 45°, and 60° using SOH CAH TOA.
• Quadrantal Angles. Find the exact trig values for 0°, 90°, 180°, 270°, and 360° using the coordinate definitions of sine, cosine, and tangent.
• Combining CAST, Reference Angles, and Special Triangles. Apply a five-step strategy to find exact trig values for any multiple of 30° or 45° in any quadrant, with four fully worked examples.
• Solving Trig Equations. Use CAST and reference angles to solve equations like sin θ = k for θ in a given domain, including domain-restriction checks.

Inside the lesson — a free preview

Why Exact Values?

Exact Values vs. Decimal Approximations. When you compute 30° on a calculator, you get 0.5 — a clean decimal. But 45°? You get something like 0.7071 which never terminates.

The exact value is (√(2))/(2) — a fraction with a radical that can be written precisely.

Exact values are NOT decimals. They are fractions and radicals. When a question asks for an exact value, leave the radical in the answer — never round to a decimal.

The 45-45-90 Triangle

Building the 45-45-90 Triangle. Start with an isosceles right triangle — a right triangle with two equal legs. Both non-right angles are 45° (since they must add to 90°).

Let both legs have length 1. Find the hypotenuse using the Pythagorean theorem:

c^2 = 1^2 + 1^2

c^2 = 2

c = √(2)

So the side lengths of a 45-45-90 triangle are 1, 1, √(2).

Exact Trig Values for 45°. Using SOH CAH TOA on the 45-45-90 triangle:

45° = (opposite)/(hypotenuse) = (1)/(√(2)) = (√(2))/(2)

45° = (adjacent)/(hypotenuse) = (1)/(√(2)) = (√(2))/(2)

45° = (opposite)/(adjacent) = (1)/(1) = 1

Note: 45° = 45° because in an isosceles right triangle the opposite and adjacent sides are equal, so their ratios with the hypotenuse come out the same.

The 30-60-90 Triangle

Deriving the 30-60-90 Triangle. Start with an equilateral triangle with all sides of length 2. All angles are 60°.

Cut this triangle in half by drawing a line from the top vertex straight down to the midpoint of the bottom side. Each half is a 30-60-90 triangle.

Finding the Long Leg. Each half-triangle has:

Hypotenuse = 2 (the original side of the equilateral triangle)

Short leg = 1 (half the original bottom side)

Find the long leg using the Pythagorean theorem:

1^2 + b^2 = 2^2

1 + b^2 = 4

b^2 = 3

b = √(3)

So the 30-60-90 triangle has sides 1, √(3), 2.

Exact Trig Values for 30° and 60°. For 30° (opposite = 1, adjacent = √(3), hypotenuse = 2):

30° = (1)/(2)

30° = (√(3))/(2)

30° = (1)/(√(3)) = (√(3))/(3) (rationalized)

For 60° (opposite = √(3), adjacent = 1, hypotenuse = 2):

60° = (√(3))/(2)

60° = (1)/(2)

60° = (√(3))/(1) = √(3)

Worked example: cos 135°

Example 1: Determine the exact value of 135°. Step 1 — Identify the quadrant.

135° is between 90° and 180°, so the terminal arm is in Quadrant II.

Step 2 — Find the reference angle.

In Quadrant II: _R = 180° - θ = 180° - 135° = 45°

Step 3 — Use CAST.

In Quadrant II, only sine is positive — so cosine is negative.

Step 4 — Look up the exact value for _R = 45°.

45° = (√(2))/(2)

Step 5 — Combine.

Answer: 135° = -(√(2))/(2)

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