MathematicsChapter 9: Trigonometric Functions
Exam Response (trigonometric Functions)
Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min.
Learning objectives
- State the main ideas and notation used in exam response (trigonometric functions).
- Apply the rules of exam response (trigonometric functions) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min. Write holes as x != formula. Amplitude is |b|, not c.
Formula
y = 1 + 2 sin(x/2). domain all reals range
Where / symbols
- EXAM RESPONSE (TRIGONOMETRIC FUNCTIONS)
- Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min. Write holes as x != formula. Amplitude is |b|, not c.
- 1. DOMAIN RANGE
- sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin: a±|b|.
- 2. GRAPH / PERIOD
- sin, cos period 2 pi/|c|. tan pi/|c|. Even cos. Odd sin. Asymptotes for tan. Middle line y=a.
Overview
Domain/range, then graph words:
Domain Range
sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin: a±|b|.
Examples csc 4x: k pi/4 · 7 cos 4x: [-7,7] · tan: all R range
Graph / Period
sin, cos period 2 pi/|c|. tan pi/|c|. Even cos. Odd sin. Asymptotes for tan. Middle line y=a.
Examples sin(2x/3): T=3 pi · 3 sin 2x: amp 3 T=pi · tan unbounded
WHAT — EXAM RESPONSE (TRIGONOMETRIC FUNCTIONS)
Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min. Write holes as x != formula. Amplitude is |b|, not c.
RULE — 1. DOMAIN RANGE
sin,cos: all R, [-1,1] then stretch. tan: holes pi/2+k pi, range R. csc: holes k pi, |y|>=1. a+b sin: a±|b|.
METHOD — 2. GRAPH / PERIOD
sin, cos period 2 pi/|c|. tan pi/|c|. Even cos. Odd sin. Asymptotes for tan. Middle line y=a.
WORKED EXAMPLE
Worked 4-mark: y = 1 + 2 sin(x/2). domain all reals range [1-2, 1+2] = [-1, 3] period 2 pi / (12) = 4 pi odd? 1+... is neither even nor odd
How to answer a exam response (trigonometric functions) 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: y = 1 + 2 sin(x/2).
- domain all reals
- range [1-2, 1+2] = [-1, 3]
- period 2 pi / (12) = 4 pi
- odd? 1+... is neither even nor odd
In simple words
A vertical shift (the +1) kills odd/even unless the shift is 0.
Key points
- period sin(cx): 2 pi/|c|
- range a+b sin: [a-|b|, a+|b|]
- holes from sin=0 or cos=0
- T then amp
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Exam Response (trigonometric Functions)
Sticky Notes · Quick Revision
Core Idea
- Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min.
- Write holes as x != formula.
- Amplitude is |b|, not c.
1. DOMAIN RANGE
- sin,cos: all R, [-1,1] then stretch.
- tan: holes pi/2+k pi, range R.
- csc: holes k pi, |y|>=1.
- a+b sin: a±|b|.
2. GRAPH / PERIOD
- sin, cos period 2 pi/|c|.
- tan pi/|c|.
- Even cos.
- Odd sin.
Worked Example
- Worked 4-mark: y = 1 + 2 sin(x/2).
- domain all reals
- range [1-2, 1+2] = [-1, 3]
- period 2 pi / (12) = 4 pi
- odd? 1+... is neither even nor odd
Common Misconception
- A vertical shift (the +1) kills odd/even unless the shift is 0.
FBISE Exam Check
- State the rule for exam response (trigonometric functions) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Typical tasks: domain of tan or csc; range of a+b sin; period of sin(cx); even/odd; sketch one period of sin or tan; max/min.
- Write holes as x != formula.
- Amplitude is |b|, not c.
- A vertical shift (the +1) kills odd/even unless the shift is 0.
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