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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.
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Domain and Range of Sin and Cos
What is it?
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.
In simple words
- 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|.
Important terms
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.
What you should learn
- 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.
Examples
- y=sin(2x)
- y=7 cos 4x
- No x to forbid
DOMAIN
No restriction for sin, cos, or a sin(cx+d). The inside cx+d can be anything. Contrast tan, which has holes (next).
RANGE
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
Recommended for Class 11
Full school-level explanation
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.
All x, then stretch:
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
DOMAIN
No restriction for sin, cos, or a sin(cx+d). The inside cx+d can be anything. Contrast tan, which has holes (next).
Remember
- all R
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.
Remember
- stretch |b|, shift a
Examples
- [-1,1]
- [-7,7]
- a+b sin: a plus minus |b|
How to answer a domain and range of sin and cos question
How it works
- 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.
Remember
- domain of sin, cos: all real
- range [-1,1]
- [-7,7]
- c does not change range of a cos(cx)
- all R
- stretch |b|, shift a
- domain of sin, cos: all real
- range [-1,1]
How the ideas connect
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
Deeper understanding
Go beyond memorising Domain and Range of Sin and Cos. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
1. DOMAIN
2. RANGE
Worked Example
Exam focus
Quick revision
- domain of sin, cos: all real
- range [-1,1]
- [-7,7]
- c does not change range of a cos(cx)
- DOMAIN AND RANGE OF SIN AND COS: sin theta and cos theta are defined for every real theta (radians in this analysis chapter). Range o
- 1. DOMAIN: No restriction for sin, cos, or a sin(cx+d). The inside cx+d can be anything. Contrast tan, which ha
The 4 in cos 4x changes the period, not the range. Range still uses |7|.
For long answers on Domain and Range of Sin and Cos: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- domain of sin, cos: all real
- range [-1,1]
- [-7,7]
- c does not change range of a cos(cx)
Important formulas
- domain of sin, cos: all real
- range [-1,1]
- [-7,7]
Common exam traps
- The 4 in cos 4x changes the period, not the range. Range still uses |7|.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The 4 in cos 4x changes the period, not the range. Range still uses |7|.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
The 4 in cos 4x changes the period, not the range. Range still uses |7|.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
domain of sin, cos: all real
range [-1,1]
[-7,7]
c does not change range of a cos(cx)
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
- 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
- Example
- y=sin(2x)
- Critical warning
- The 4 in cos 4x changes the period, not the range. Range still uses |7|.
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Exam Practice
Define
Q1. Define Domain and Range of Sin and Cos.
Explain
Q2. Explain: DOMAIN.
Example
Q3. Give one clear example of Domain and Range of Sin and Cos and say why it fits.
Mistake
Q4. State one common mistake students make about Domain and Range of Sin and Cos.
Exam note
Q5. Write a short exam note on Domain and Range of Sin and Cos: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.