MathematicsChapter 8: Fundamentals of Trigonometry

Exam Response (fundamentals of Trigonometry)

Typical tasks: expand sin(A+B); tan(A+B); allied reduce; product-to-sum; sum-to-product; a numerical 15/75 angle.

Learning objectives

State the main ideas and notation used in exam response (fundamentals of trigonometry).
Apply the rules of exam response (fundamentals of trigonometry) in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

Typical tasks: expand sin(A+B); tan(A+B); allied reduce; product-to-sum; sum-to-product; a numerical 1575 angle. Quote the exact identity the question needs. Check even/odd for negatives.

Mathematical form

A=45 B

sin30 = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(12

Where / symbols

EXAM RESPONSE (FUNDAMENTALS OF TRIGONOMETRY): Typical tasks: expand sin(A+B); tan(A+B); allied reduce; product-to-sum; sum-to-product; a numerical 15/75 angle. Quote the exact identity the question needs. Check even/odd for negatives.
1. COMPOUND: Write the full four-term or tan fraction. 75=45+30, 15=45-30. Double if 2A appears.
2. ALLIED AND SUM/PRODUCT: Reduce to 30/45/60. Product 2 sin cos. Sum 2 sin(avg) cos(half). Cos-cos minus.

Overview

Expand, then reduce:

Compound

Write the full four-term or tan fraction. 75=45+30, 15=45-30. Double if 2A appears.

Examples sin(A+B) both + · tan bottom 1-st · cos 2A three forms

Allied and Sum/product

Reduce to 30/45/60. Product 2 sin cos. Sum 2 sin(avg) cos(half). Cos-cos minus.

Examples 180-A sin+ · 2 sin cos · cos-cos has -2

WHAT — EXAM RESPONSE (FUNDAMENTALS OF TRIGONOMETRY)

Typical tasks: expand sin(A+B); tan(A+B); allied reduce; product-to-sum; sum-to-product; a numerical 1575 angle. Quote the exact identity the question needs. Check even/odd for negatives.

RULE — 1. COMPOUND

Write the full four-term or tan fraction. 75=45+30, 15=45-30. Double if 2A appears.

METHOD — 2. ALLIED AND SUM/PRODUCT

Reduce to 30/45/60. Product 2 sin cos. Sum 2 sin(avg) cos(half). Cos-cos minus.

WORKED EXAMPLE

Worked 4-mark cos(A+B) with A=45 B=30. cos45 cos30 - sin45 sin30 = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(12) = sqrt(2)/2 * (sqrt(3)-1)/2 = (sqrt(6)-sqrt(2))/4

How to answer a exam response (fundamentals of trigonometry) 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

Worked 4-mark cos(A+B) with A=45 B=30.
cos45 cos30 - sin45 sin30
= (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(12)
= sqrt(2)/2 * (sqrt(3)-1)/2
= (sqrt(6)-sqrt(2))/4

In simple words

Minus between the products for cos(A+B). Plus would be cos(A-B).

Key points

compound: mixed products
CAST for allied signs
write every product
CAST + table

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.