MathematicsChapter 8: Fundamentals of Trigonometry

More Allied Forms (270, Negative)

sin(360-A)=-sin A, cos(360-A)=cos A.

Learning objectives

State the main ideas and notation used in more allied forms (270, negative).
Apply the rules of more allied forms (270, negative) in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST covers almost every allied question. Learn a table or derive from 180 and 90.

Where / symbols

MORE ALLIED FORMS (270, NEGATIVE): sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST covers almost every allied question. Learn a table or derive from 180 and 90.
1. NEGATIVE ANGLES: sin(-30)=-1/2. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar form too.
2. 270 AND 360: 270=180+90, so use two steps if you forget the table. 360-A: like -A for sine and cosine of a positive A between 0 and 360.

Overview

Even/odd, then 270:

Negative Angles

sin(-30)=-12. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar form too.

Examples sin(-A)=-sin A · cos(-A)=cos A · tan odd

270 and 360

270=180+90, so use two steps if you forget the table. 360-A: like -A for sine and cosine of a positive A between 0 and 360.

Examples sin(270-A)=-cos A school table · cos(360-A)=cos A · Do not mix with 180-A

WHAT — MORE ALLIED FORMS (270, NEGATIVE)

RULE — 1. NEGATIVE ANGLES

sin(-30)=-12. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar form too.

METHOD — 2. 270 AND 360

270=180+90, so use two steps if you forget the table. 360-A: like -A for sine and cosine of a positive A between 0 and 360.

WORKED EXAMPLE

Worked: cos(-60). cos even: cos(-60)=cos 60 cos 60 = 12 not -12 (that would be sine)

How to answer a more allied forms (270, negative) 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

cos(-60).
cos even: cos(-60)=cos 60
cos 60 = 12
not -12 (that would be sine)

In simple words

Cosine is even: negatives do not flip cos. Sine does flip.

Key points

sin odd, cos even
360-A is a fourth-quadrant picture
odd / even
two-step if needed

Exam preparation

Exam Focus

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