MathematicsChapter 9: Trigonometric Functions
Domain and Range of Sin and Cos
sin theta and cos theta are defined for every real theta (radians in this analysis chapter).
Learning objectives
- State the main ideas and notation used in domain and range of sin and cos.
- Apply the rules of domain and range of sin and cos in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
sin theta and cos theta are defined for every real theta (radians in this analysis chapter). Range of both is [-1, 1]. y = a sin theta has range [-|a|, |a|]. y = 7 cos 4x still has domain all reals and range [-7,7]. Meaning: sine and cosine never blow up; they only wobble between -1 and 1, then you stretch by a.
Mathematical form
y = a sin theta has range
y = 7 cos 4x still has domain all reals and
y = 7 cos 4x. -1
Where / symbols
- DOMAIN AND RANGE OF SIN AND COS
- sin theta and cos theta are defined for every real theta (radians in this analysis chapter). Range of both is [-1, 1]. y = a sin theta has range [-|a|, |a|]. y = 7 cos 4x still has domain all reals and range [-7,7]. Meaning: sine and cosine never blow up; they only wobble between -1 and 1, then you stretch by a.
- 1. DOMAIN
- No restriction for sin, cos, or a sin(cx+d). The inside cx+d can be anything. Contrast tan, which has holes (next).
- 2. RANGE
- -1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
Overview
All x, then stretch:
Domain
No restriction for sin, cos, or a sin(cx+d). The inside cx+d can be anything. Contrast tan, which has holes (next).
Examples y=sin(2x) · y=7 cos 4x · No x to forbid
Range
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
Examples [-1,1] · [-7,7] · a+b sin: a plus minus |b|
WHAT — DOMAIN AND RANGE OF SIN AND COS
sin theta and cos theta are defined for every real theta (radians in this analysis chapter). Range of both is [-1, 1]. y = a sin theta has range [-|a|, |a|]. y = 7 cos 4x still has domain all reals and range [-7,7]. Meaning: sine and cosine never blow up; they only wobble between -1 and 1, then you stretch by a.
RULE — 1. DOMAIN
No restriction for sin, cos, or a sin(cx+d). The inside cx+d can be anything. Contrast tan, which has holes (next).
METHOD — 2. RANGE
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
WORKED EXAMPLE
Worked: range of y = 7 cos 4x. -1 <= cos 4x <= 1 for all real x -7 <= 7 cos 4x <= 7 range [-7, 7] domain all real x
How to answer a domain and range of sin and cos 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
- range of y = 7 cos 4x.
- -1 <= cos 4x <= 1 for all real x
- -7 <= 7 cos 4x <= 7
- range [-7, 7]
- domain all real x
In simple words
The 4 in cos 4x changes the period, not the range. Range still uses |7|.
Key points
- domain of sin, cos: all real
- range [-1,1]
- all R
- stretch |b|, shift a
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Domain and Range of Sin and Cos
Sticky Notes · Quick Revision
Core Idea
- sin theta and cos theta are defined for every real theta (radians in this analysis chapter).
- Range of both is [-1, 1].
- y = a sin theta has range [-|a|, |a|].
1. DOMAIN
- No restriction for sin, cos, or a sin(cx+d).
- The inside cx+d can be anything.
- Contrast tan, which has holes (next).
2. RANGE
- -1 <= sin(anything) <= 1.
- Multiply by 7: -7 to 7.
- a + b sin theta: [a-|b|, a+|b|].
- Shift then stretch.
Worked Example
- Worked: range of y = 7 cos 4x.
- -1 <= cos 4x <= 1 for all real x
- -7 <= 7 cos 4x <= 7
- range [-7, 7]
- domain all real x
Common Misconception
- The 4 in cos 4x changes the period, not the range. Range still uses |7|.
FBISE Exam Check
- State the rule for domain and range of sin and cos before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- sin theta and cos theta are defined for every real theta (radians in this analysis chapter).
- Range of both is [-1, 1].
- y = a sin theta has range [-|a|, |a|].
- The 4 in cos 4x changes the period, not the range. Range still uses |7|.
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