The Special Triangles

Exact trigonometric values come from two special right-angled triangles. These values must be memorised for Year 11 and HSC examinations — calculators give decimal approximations, but exact values are required.

45°-45°-90° Triangle

1

√2

Sides in ratio 1 : 1 : √2

30°-60°-90° Triangle

√3

1

2

Sides in ratio 1 : √3 : 2

Table of Exact Values

The following table summarises the exact values of sin, cos and tan for the five key angles. You must memorise these values.

Angle	0°	30°	45°	60°	90°
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	1/√3	1	√3	undefined

Memory tip: For sin, write √0/2, √1/2, √2/2, √3/2, √4/2 which gives 0, ½, √2/2, √3/2, 1. Cosine values are the reverse order.

Using Exact Values in Calculations

In HSC Mathematics, you will frequently need to evaluate expressions involving exact trig values. The key is to substitute the exact values and simplify using surd arithmetic.

1

Rationalise denominators: If the answer contains 1/√3, multiply top and bottom by √3 to get √3/3.

2

Combine like surds: √2 + 3√2 = 4√2. Only surds with the same radicand can be added.

3

Use identities: sin²θ + cos²θ = 1 can simplify expressions. For 45°: (√2/2)² + (√2/2)² = ½ + ½ = 1.

Key Vocabulary

Exact Value

A trigonometric value expressed using surds and fractions, not decimal approximations.

Special Triangle

The 30-60-90 and 45-45-90 right-angled triangles used to derive exact trig values.

Rationalise

Rewriting a fraction so the denominator contains no surds (e.g., multiply by √3/√3).

Surd

An irrational root expression such as √2 or √3 that cannot be simplified to a rational number.

Worked Examples

1

Evaluate sin²(60°) + cos²(60°) without a calculator.

Step 1: sin(60°) = √3/2, so sin²(60°) = (√3/2)² = 3/4

Step 2: cos(60°) = 1/2, so cos²(60°) = (1/2)² = 1/4

Answer: 3/4 + 1/4 = 1 (confirming the Pythagorean identity)

2

Find the exact value of 2sin(30°)cos(30°).

Step 1: sin(30°) = 1/2 and cos(30°) = √3/2

Step 2: 2 × 1/2 × √3/2 = 2√3/4 = √3/2

Answer: 2sin(30°)cos(30°) = √3/2 (which equals sin(60°), demonstrating the double angle formula)

3

Rationalise and simplify: 1/tan(60°)

Step 1: tan(60°) = √3, so the expression is 1/√3

Step 2: Rationalise: 1/√3 × √3/√3 = √3/3

Answer: √3/3 (which also equals tan(30°))

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What is the exact value of sin(45°)?

√2/2 1/2 √3/2 1

Question 2

What is the exact value of tan(30°)?

√3 1/√3 1 √3/2

Question 3

What is the exact value of cos(60°)?

√3/2 √2/2 1/2 0

Question 4

What is the exact value of sin(30°) + cos(60°)?

1/2 √3/2 √3 1

Question 5

Why is tan(90°) undefined?

Because cos(90°) = 0, and tan = sin/cos involves dividing by zero Because sin(90°) = 0 Because the triangle has no hypotenuse at 90° Because 90° is not a real angle

Trigonometric Exact Values