Mathematics Advanced • Year 11 • Module 2 • Lesson 5

Exact Values & Special Triangles

Build procedural fluency in exact trig values at 30°, 45°, 60° (and their related angles in all four quadrants), including rationalising denominators.

Build · Skill Drill

1. Quick recall

Fill in the exact-value table below from memory. 1 mark each row

Q1.1 The exact-value table for Quadrant I:

θ	sin θ	cos θ	tan θ
0
30° = π/6
45° = π/4
60° = π/3
90° = π/2

Q1.2 Rationalise: 1/√2 = ____________ and 1/√3 = ____________

Q1.3 The 30-60-90 special triangle has sides in the ratio ____ : ____ : ____. The 45-45-90 special triangle has sides in the ratio ____ : ____ : ____.

Stuck? Revisit lesson § Exact values from special triangles.

2. Worked example — exact value at 150°

Follow each line of algebra. Every step has a reason on the right.

Problem. Find the exact value of cos 150°.

Step 1 — Identify the quadrant.

90° < 150° < 180° ⇒ Quadrant II.

Reason: needed for ASTC sign rule in Step 4.

Step 2 — Find the reference angle.

α = 180° − 150° = 30°.

Reason: in QII, reference angle = 180° − θ.

Step 3 — Look up the exact value at the reference angle.

cos 30° = √3/2 (from the 30-60-90 triangle).

Step 4 — Apply the ASTC sign.

In QII, only sine is positive ⇒ cosine is negative.

cos 150° = −cos 30° = −√3/2.

Conclusion. cos 150° = −√3/2.

3. Faded example — exact value at 4π/3

Find the exact value of tan(4π/3). Fill in each blank line. 4 marks

Step 1 — Convert to degrees for clarity (optional).

4π/3 × (180/π) = ________°.

Step 2 — Identify the quadrant.

________° is between ________° and ________°, so Quadrant ________.

Step 3 — Find the reference angle.

In QIII, α = θ − 180° = ________° − 180° = ________° (i.e. π/____ rad).

Step 4 — Look up the exact value at the reference angle.

tan 60° = ____________.

Step 5 — Apply ASTC.

In QIII, tangent is ________________ (positive / negative). So tan(4π/3) = ____________.

Conclusion. tan(4π/3) = ____________________.

Stuck? Revisit lesson § Worked Example 2 — Quadrant III.

4. Graduated practice — exact values

Find each exact value without a calculator. Rationalise denominators. Show the reference angle and ASTC sign reasoning.

Foundation — QI / boundary angles (4 questions)

Q	Find exact value of	Answer (exact, rationalised)
4.1 1	sin 30°
4.2 1	cos 60°
4.3 1	tan 45°
4.4 1	tan 30°

Standard — reference angle + ASTC (6 questions)

State quadrant and reference angle before quoting the exact value.

4.5 sin 120°. 2 marks

4.6 cos(5π/4). 2 marks

4.7 tan 300°. 2 marks

4.8 sin(7π/6). 2 marks

4.9 cos(11π/6). 2 marks

4.10 Simplify sin²(45°) + cos²(45°) and explain in one line why this equals 1 by direct computation. 2 marks

Extension — combine concepts (2 questions)

4.11 Find the exact value of sin 135° + cos 135°. Show your working clearly. 3 marks

4.12 Without a calculator, show that tan 60° − tan 30° = 2√3/3. (Rationalise any denominators before subtracting.) 3 marks

Stuck on 4.12? Substitute tan 60° = √3 and tan 30° = √3/3 (rationalised from 1/√3), find a common denominator, then subtract.

5. Self-check the easy 3

Tick the first three once you've verified your method works.

For 4.1 (sin 30°) I got 1/2 — not √3/2 (Trap 1 in the lesson: don't swap 30 and 60).

For 4.4 (tan 30°) I rationalised: 1/√3 = √3/3 (Trap 2 in the lesson: don't leave un-rationalised).

For 4.5 (sin 120°) I got +√3/2 and applied ASTC explicitly (QII, sine positive).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — QI exact-value table

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

Q1.2 — Rationalised forms

1/√2 = √2/2. 1/√3 = √3/3. (Multiply numerator and denominator by the surd.)

Q1.3 — Special-triangle side ratios

30-60-90 triangle: 1 : √3 : 2 (shortest opposite 30°, then opposite 60°, then hypotenuse).
45-45-90 triangle: 1 : 1 : √2 (the two legs are equal, hypotenuse is √2 times the leg).

Q3 — Faded example: tan(4π/3)

Step 1: 4π/3 × (180/π) = 240°.
Step 2: 240° is between 180° and 270°, so Quadrant III.
Step 3: In QIII, α = 240° − 180° = 60° (i.e. π/3 rad).
Step 4: tan 60° = √3.
Step 5: In QIII, tangent is positive (both sin and cos negative, ratio positive). So tan(4π/3) = √3.
Conclusion: tan(4π/3) = √3.

Q4.1 — sin 30°

1/2. (From the 30-60-90 triangle: shortest side / hypotenuse = 1/2.)

Q4.2 — cos 60°

1/2. (From the 30-60-90 triangle: complementary, so cos 60° = sin 30° = 1/2.)

Q4.3 — tan 45°

1. (In the 45-45-90 triangle, opposite = adjacent.)

Q4.4 — tan 30°

tan 30° = sin 30° / cos 30° = (1/2) / (√3/2) = 1/√3 = √3/3 (rationalised).

Q4.5 — sin 120°

QII; ref = 180° − 120° = 60°; sin 60° = √3/2; QII sine positive. sin 120° = √3/2.

Q4.6 — cos(5π/4)

5π/4 = 225° in QIII; ref = 225° − 180° = 45°; cos 45° = √2/2; QIII cosine negative. cos(5π/4) = −√2/2.

Q4.7 — tan 300°

QIV; ref = 360° − 300° = 60°; tan 60° = √3; QIV tangent negative. tan 300° = −√3.

Q4.8 — sin(7π/6)

7π/6 = 210° in QIII; ref = 210° − 180° = 30°; sin 30° = 1/2; QIII sine negative. sin(7π/6) = −1/2.

Q4.9 — cos(11π/6)

11π/6 = 330° in QIV; ref = 360° − 330° = 30°; cos 30° = √3/2; QIV cosine positive. cos(11π/6) = √3/2.

Q4.10 — sin²(45°) + cos²(45°)

= (√2/2)² + (√2/2)² = 2/4 + 2/4 = 1/2 + 1/2 = 1. This is the Pythagorean identity, verified explicitly using the exact 45° values.

Q4.11 — sin 135° + cos 135°

135° is in QII, ref = 180° − 135° = 45°. sin 135° = +sin 45° = √2/2 (QII sine positive). cos 135° = −cos 45° = −√2/2 (QII cosine negative).
Sum: √2/2 − √2/2 = 0.

Q4.12 — Show tan 60° − tan 30° = 2√3/3

tan 60° = √3; tan 30° = 1/√3 = √3/3 (rationalised).
tan 60° − tan 30° = √3 − √3/3.
Common denominator 3: √3 = 3√3/3. So expression = 3√3/3 − √3/3 = 2√3/3. ✓ (Hence shown.)