Allied Angles

Allied Angles

What is it?

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.

In simple words

• 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.

Important terms

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.

What you should learn

• 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.

Examples

• sin(90-A)=cos A
• sec(90-A)=csc A
• Same idea pi/2 - A

90 MINUS

sin(90-A)=cos A, cos(90-A)=sin A, tan(90-A)=cot A. Complementary. Use to rewrite.

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.

Full school-level explanation

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.

Co-function, then 180:

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

90 MINUS

sin(90-A)=cos A, cos(90-A)=sin A, tan(90-A)=cot A. Complementary. Use to rewrite.

Remember

• complement

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.

Remember

• ASTC / CAST

Examples

• 180-A: sin +
• 180+A: both sin and cos -
• 360-A: cos +

How to answer a allied angles question

How it works

• 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.

Remember

• reduce to A, then sign from the quadrant
• sin(90-A)=cos A
• 12
• sine + in Q2
• complement
• ASTC / CAST
• reduce to A, then sign from the quadrant
• sin(90-A)=cos A

How the ideas connect

Worked: sin 150. 150 = 180-30 sin(180-30)=sin 30 sin 30 = 12

Deeper understanding

Go beyond memorising Allied Angles. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: sin 150. 150 = 180-30 sin(180-30)=sin 30 sin 30 = 12

Common mistakes

Exam focus

Core Idea

1. 90 MINUS

2. 180 AND CAST

Worked Example

Exam focus

Quick revision

• reduce to A, then sign from the quadrant
• sin(90-A)=cos A
• 12
• sine + in Q2
• ALLIED ANGLES: Allied angles are built from A by adding or subtracting 90, 180, 270, 360 (and negatives). Examples:
• 1. 90 MINUS: sin(90-A)=cos A, cos(90-A)=sin A, tan(90-A)=cot A. Complementary. Use to rewrite.

sin(180-A)=+sin A, but cos(180-A)=-cos A. Do not give both a plus.

For long answers on Allied Angles: define clearly → explain the main parts → give one example → state one exam caution.