MathematicsChapter 8: Fundamentals of Trigonometry

Tan(a±b) and Double Angle

tan(A+B)=(tan A + tan B)/(1 - tan A tan B), provided the cosine in the bottom is not zero (defined).

Learning objectives

State the main ideas and notation used in tan(a±b) and double angle.
Apply the rules of tan(a±b) and double angle in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

tan(A+B)=tan A + tan B1 - tan A tan B, provided the cosine in the bottom is not zero (defined). tan(A-B)=tan A - tan B1 + tan A tan B. Double: sin 2A = 2 sin A cos A, cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A, tan 2A = 2 tan A1-tan² A. Meaning: divide sin compound by cos compound and divide through by cos A cos B.

Mathematical form

tan45=1, tan30=1/sqrt(3)

tan75 = 1 + 1/sqrt(3)1 - 1/sqrt(3)

A = 2 cos² A - 1

tan45=1

Where / symbols

TAN(A±B) AND DOUBLE ANGLE: tan(A+B)=(tan A + tan B)/(1 - tan A tan B), provided the cosine in the bottom is not zero (defined). tan(A-B)=(tan A - tan B)/(1 + tan A tan B). Double: sin 2A = 2 sin A cos A, cos 2A = cos^2 A - sin^2 A = 2 cos^2 A - 1 = 1 - 2 sin^2 A, tan 2A = 2 tan A / (1-tan^2 A). Meaning: divide sin compound by cos compound and divide through by cos A cos B.
1. TAN SUM: Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined when that is 0 (A+B odd multiple of 90 degrees school).
2. DOUBLE: Put B=A. Three forms of cos 2A are all useful (pick the one that matches sin or cos on the paper). Half-angle later if asked via rearranging cos 2A.

Relationship

tan(A+B)=s+t1-st
sin 2A = 2 sin A cos A

Overview

Divide sin/cos, then double:

Tan Sum

Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined when that is 0 (A+B odd multiple of 90 degrees school).

Examples 1 - product in bottom · Defined when cos(A+B)!=0 · tan(A-B) plus in bottom

Double

Put B=A. Three forms of cos 2A are all useful (pick the one that matches sin or cos on the paper). Half-angle later if asked via rearranging cos 2A.

Examples 2 sin cos · cos^2-sin^2 · 2t/(1-t^2) for tan

WHAT — TAN(A±B) AND DOUBLE ANGLE

RULE — 1. TAN SUM

Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined when that is 0 (A+B odd multiple of 90 degrees school).

METHOD — 2. DOUBLE

Put B=A. Three forms of cos 2A are all useful (pick the one that matches sin or cos on the paper). Half-angle later if asked via rearranging cos 2A.

WORKED EXAMPLE

Worked: tan 75 = tan(45+30). tan45=1, tan30=1/sqrt(3) tan75 = 1 + 1/sqrt(3)1 - 1/sqrt(3) times sqrt(3)/sqrt(3): sqrt(3)+1sqrt(3)-1

How to answer a tan(a±b) and double angle 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

tan 75 = tan(45+30).
tan45=1, tan30=1/sqrt(3)
tan75 = 1 + 1/sqrt(3)1 - 1/sqrt(3)
times sqrt(3)/sqrt(3): sqrt(3)+1sqrt(3)-1

In simple words

Bottom 1 - tan A tan B. If you write 1 + by accident, A+B is wrong.

Key points

tan(A+B)=s+t1-st
sin 2A = 2 sin A cos A
divide by cos cos
B=A

Exam preparation

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