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Study GuideYear 11Level 3 · ChallengeHSCVCE

Free Year 11 Trigonometric Functions, Radians and Exact Values

An HSC/VCE-style study guide to radian measure, the exact-value table, the unit circle (ASTC) and the basic trigonometric graphs. Worked examples convert between degrees and radians, find exact values of angles in any quadrant, calculate arc length and sector area, solve trigonometric equations on a given domain and read amplitude and period from an equation, with a six-question self-check and answers.

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Year
Year 11
Subject
Mathematics
Topic
Trigonometry
Difficulty
Level 3 · Challenge
Estimated time
30 minutes
Curriculum
Australian Curriculum
Answers
Not applicable
Format
PDF (A4) + print

Students will practise

  • converting between degrees and radians
  • finding exact values of sin, cos and tan for angles in any quadrant using the unit circle
  • calculating arc length and sector area using radians
  • solving trigonometric equations on a given domain and identifying amplitude and period

Curriculum: Australian Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Trigonometric Functions, Radians and Exact Values

  1. 1Year 11 Trigonometry (Finding Sides) Worksheet — Level 3
Study guide

How to use this study guide

  1. Read it together first, pausing at each worked example to try the step before reading the answer.
  2. Attempt the "Check yourself" questions at the end without looking back.
  3. Then practise with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this study guide for?

Year 11 students (typically ages 16–17) working on trigonometry. It is pitched at level 3 · challenge.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 30 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try Year 11 Trigonometry (Finding Sides) Worksheet — Level 3.

Preview

Year 11 · Mathematics · Trigonometry

Trigonometric Functions, Radians and Exact Values

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What you need to know

A radian is the angle made at the centre of a circle by an arc the same length as the radius. A full turn is 2π radians, so π radians = 180°. To convert degrees to radians multiply by ; to convert radians to degrees multiply by . Radians are used in calculus and in the formulas arc length l = rθ and sector area A = [frac 1/2]r[sup 2]θ, which only work when θ is in radians.

Angle0 (0°) (30°) (45°) (60°) (90°)
sin01
cos10
tan01√3undefined

For angles beyond 90° use the unit circle. Measuring anticlockwise from the positive x-axis, the four quadrants are remembered by ASTC ("All Stations To Central"): in quadrant 1 All ratios are positive; in quadrant 2 only Sin is positive; in quadrant 3 only Tan; in quadrant 4 only Cos. To find an exact value: (1) decide the quadrant, (2) find the reference angle — the acute angle made with the x-axis (180° − θ, θ − 180° or 360° − θ), (3) take the exact value of the reference angle and attach the sign from ASTC.

Convert 150° to radians and to degrees

  1. 150° × = = [frac 5π/6]
  2. × = = 135°

Find the exact values of sin 150°, cos 210° and tan

  1. sin 150°: quadrant 2 (sin positive). Reference angle 180° − 150° = 30°. sin 150° = sin 30° = [frac 1/2]
  2. cos 210°: quadrant 3 (cos negative). Reference angle 210° − 180° = 30°. cos 210° = −cos 30° = −[frac √3/2]
  3. tan : = 225°, quadrant 3 (tan positive). Reference angle − π = . tan = tan = 1

A sector has radius 6 cm and angle . Find the arc length and the area.

  1. Arc length: l = rθ = 6 × = 2π cm ≈ 6.28 cm
  2. Area: A = r2θ = × 36 × = 6π cm[sup 2] ≈ 18.85 cm[sup 2]

Solve sin x = for 0 ≤ x ≤ 2π

  1. sin is positive in quadrants 1 and 2.
  2. Reference angle: sin−1() = .
  3. Quadrant 1: x = . Quadrant 2: x = π − = .
  4. Solutions: x = [frac π/6], [frac 5π/6]

Solve 2 cos x + √3 = 0 for 0 ≤ x ≤ 2π

  1. Rearrange: cos x = −.
  2. cos is negative in quadrants 2 and 3. Reference angle: cos−1() = .
  3. Quadrant 2: x = π − = . Quadrant 3: x = π + = .
  4. Solutions: x = [frac 5π/6], [frac 7π/6]

State the amplitude and period of y = 3 sin 2x

  1. For y = a sin (nx) or y = a cos (nx): amplitude = |a| and period = .
  2. Amplitude = 3 (the graph oscillates between −3 and 3).
  3. Period = = π (one full wave every π units, so there are two full waves between 0 and 2π).
Common mistake: a calculator in the wrong mode. If a question uses radians (any angle written with π, or an arc-length or calculus question), the calculator must be in radian mode. Test: sin should give 0.5. Another common mistake: writing a decimal when an exact value is required. "Exact" means leave surds and π in the answer: sin 60° = , not 0.87. Also remember that ASTC gives the sign only — the size always comes from the reference angle.

Check yourself

  1. Convert 60° to radians.
  2. Convert to degrees.
  3. Find the exact value of cos 120°.
  4. Find the exact value of sin .
  5. A sector has radius 10 cm and angle . Find the arc length.
  6. Solve tan x = 1 for 0 ≤ x ≤ 2π.

Answers: 1. 2. 210° 3. − (quadrant 2, reference angle 60°) 4. − (quadrant 3, reference angle ) 5. 10 × = 2π cm 6. x = ,

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Created by
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Last reviewed
1 October 2026
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