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Year 11 Maths

Advanced Trigonometry

Master the unit circle, exact trigonometric values, and solve trigonometric equations across all four quadrants.

The Unit Circle

The unit circle is a circle with radius 1, centred at the origin. For an angle θ measured from the positive x-axis, the point on the circle is (cosθ, sinθ).

cos2θ + sin2θ = 1

The Pythagorean identity — true for all angles.

Quadrant Signs (ASTC)

Q2: Sin +
Q1: All +
Q3: Tan +
Q4: Cos +

tanθ

tanθ = sinθ / cosθ

Undefined when cosθ = 0 (at 90° and 270°).

Exact Trigonometric Values

You must memorise the exact values for the special angles: 0°, 30°, 45°, 60°, and 90°.

Angle 0° 30° 45° 60° 90°
sin 0 1/2 √2/2 √3/2 1
cos 1 √3/2 √2/2 1/2 0
tan 0 √3/3 1 √3 undef.

Solving Trigonometric Equations

When solving trig equations, find the reference angle first, then determine which quadrants give solutions based on the sign of the trig function.

1

Example: Solve sinθ = 1/2 for 0° ≤ θ ≤ 360°

Step 1: Reference angle: sin−1(1/2) = 30°

Step 2: sin is positive in Q1 and Q2.

Step 3: Q1: θ = 30°. Q2: θ = 180° − 30° = 150°.

Solutions: θ = 30° and 150°

2

Example: Solve cosθ = −√3/2 for 0° ≤ θ ≤ 360°

Step 1: Reference angle: cos−1(√3/2) = 30°

Step 2: cos is negative in Q2 and Q3.

Step 3: Q2: θ = 180° − 30° = 150°. Q3: θ = 180° + 30° = 210°.

Solutions: θ = 150° and 210°

Knowledge Check

Test your understanding of advanced trigonometry.

Question 1

What is the exact value of sin 60°?

Question 2

In which quadrants is cosθ negative?

Question 3

What is tan 45°?

Question 4

Solve: cosθ = 0 for 0° ≤ θ ≤ 360°

Question 5

If sinθ = √2/2 and θ is in Q2, what is θ?

Question 6

Simplify: sin2θ + cos2θ

Question 7

Solve: tanθ = −1 for 0° ≤ θ ≤ 360°

Question 8

What is the exact value of cos 150°?

Question 9

If sinθ = 3/5 and θ is in Q1, find cosθ.

Question 10

Solve: 2sinθ − 1 = 0 for 0° ≤ θ ≤ 360°

Key Concepts Summary