MathematicsChapter 8: Fundamentals of Trigonometry
Allied Angles
Allied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives).
Learning objectives
- State the main ideas and notation used in allied angles.
- Apply the rules of allied angles in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Allied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives). Examples: sin(180-A)=sin A, cos(180-A)=-cos A, tan(180-A)=-tan A. sin(90-A)=cos A. The CAST (or ASTC) rule names which functions are positive in which quadrant. Meaning: reduce to an acute related angle, then attach the right sign.
Where / symbols
- ALLIED ANGLES
- Allied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives). Examples: sin(180-A)=sin A, cos(180-A)=-cos A, tan(180-A)=-tan A. sin(90-A)=cos A. The CAST (or ASTC) rule names which functions are positive in which quadrant. Meaning: reduce to an acute related angle, then attach the right sign.
- 1. 90 MINUS
- sin(90-A)=cos A, cos(90-A)=sin A, tan(90-A)=cot A. Complementary. Use to rewrite.
- 2. 180 AND CAST
- Second quadrant: sine +. sin(180-A)=+sin A. cos(180-A)=-cos A. Third: tan +. Fourth: cos +. Draw a quick CAST if needed.
Relationship
- sin(90-A)=cos A
Overview
Co-function, then 180:
90 Minus
sin(90-A)=cos A, cos(90-A)=sin A, tan(90-A)=cot A. Complementary. Use to rewrite.
Examples sin(90-A)=cos A · sec(90-A)=csc A · Same idea pi/2 - A
180 and Cast
Second quadrant: sine +. sin(180-A)=+sin A. cos(180-A)=-cos A. Third: tan +. Fourth: cos +. Draw a quick CAST if needed.
Examples 180-A: sin + · 180+A: both sin and cos - · 360-A: cos +
WHAT — ALLIED ANGLES
Allied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives). Examples: sin(180-A)=sin A, cos(180-A)=-cos A, tan(180-A)=-tan A. sin(90-A)=cos A. The CAST (or ASTC) rule names which functions are positive in which quadrant. Meaning: reduce to an acute related angle, then attach the right sign.
RULE — 1. 90 MINUS
sin(90-A)=cos A, cos(90-A)=sin A, tan(90-A)=cot A. Complementary. Use to rewrite.
METHOD — 2. 180 AND CAST
Second quadrant: sine +. sin(180-A)=+sin A. cos(180-A)=-cos A. Third: tan +. Fourth: cos +. Draw a quick CAST if needed.
WORKED EXAMPLE
Worked: sin 150. 150 = 180-30 sin(180-30)=sin 30 sin 30 = 12
How to answer a allied angles question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- sin 150.
- 150 = 180-30
- sin(180-30)=sin 30
- sin 30 = 12
In simple words
sin(180-A)=+sin A, but cos(180-A)=-cos A. Do not give both a plus.
Key points
- reduce to A, then sign from the quadrant
- sin(90-A)=cos A
- complement
- ASTC / CAST
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Allied Angles
Sticky Notes · Quick Revision
Core Idea
- Allied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives).
- Examples: sin(180-A)=sin A, cos(180-A)=-cos A, tan(180-A)=-tan A.
- sin(90-A)=cos A.
1. 90 MINUS
- sin(90-A)=cos A, cos(90-A)=sin A, tan(90-A)=cot A.
- Complementary.
- Use to rewrite.
2. 180 AND CAST
- Second quadrant: sine +.
- sin(180-A)=+sin A.
- cos(180-A)=-cos A.
- Third: tan +.
Worked Example
- Worked: sin 150.
- 150 = 180-30
- sin(180-30)=sin 30
- sin 30 = 12
Common Misconception
- sin(180-A)=+sin A, but cos(180-A)=-cos A. Do not give both a plus.
FBISE Exam Check
- State the rule for allied angles before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Allied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives).
- Examples: sin(180-A)=sin A, cos(180-A)=-cos A, tan(180-A)=-tan A.
- sin(90-A)=cos A.
- sin(180-A)=+sin A, but cos(180-A)=-cos A. Do not give both a plus.
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