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Range of Sine and Cosine
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
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Range of Sine and Cosine
What is it?
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
In simple words
- -1 <= sin(anything) <= 1.
- [-1,1]
- [-7,7]
- State the rule, show the working, and check the answer.
Important terms
RANGE OF SINE AND COSINE: -1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
RANGE: -1 <= sin(anything) <= 1.
What you should learn
- State the rule or meaning of range of sine and cosine in easy words.
- Apply range of sine and cosine in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Examples
- [-1,1]
- [-7,7]
- a+b sin: a plus minus |b|
MAIN IDEA
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
EXAM USE
Write the rule first: -1 <= sin(anything) <= 1. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Recommended for Class 11
Full school-level explanation
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
Understand the idea, then apply it:
WHAT — RANGE OF SINE AND COSINE
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
WHY — Why it matters
This rule is a separate learning unit inside domain and range of sin and cos. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: -1 <= sin(anything) <= 1. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
MAIN IDEA
-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|
EXAM USE
Write the rule first: -1 <= sin(anything) <= 1. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Remember
- focused answer
- accurate support
Examples
- State the central idea
- Use one accurate detail
- Check the final response
How to answer a range of sine and cosine question
How it works
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
Remember
- stretch |b|, shift a
- domain of sin, cos: all real
- range [-1,1]
- c does not change range of a cos(cx)
- stretch |b|, shift a
- focused answer
- accurate support
- stretch |b|, shift a
Deeper understanding
Go beyond memorising Range of Sine and Cosine. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
What to Know
Example or Evidence
How to Answer
Exam focus
Quick revision
- stretch |b|, shift a
- domain of sin, cos: all real
- range [-1,1]
- c does not change range of a cos(cx)
- RANGE OF SINE AND COSINE: -1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretc
- RANGE: -1 <= sin(anything) <= 1.
The 4 in cos 4x changes the period, not the range. Range still uses |7|.
For long answers on Range of Sine and Cosine: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- stretch |b|, shift a
- domain of sin, cos: all real
- range [-1,1]
- c does not change range of a cos(cx)
Important formulas
- stretch |b|, shift a
- domain of sin, cos: all real
- range [-1,1]
Common exam traps
- The 4 in cos 4x changes the period, not the range. Range still uses |7|.
- Do not choose a 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 choose a formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes, restrictions, or skipped algebra steps.
- 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.
stretch |b|, shift a
domain of sin, cos: all real
range [-1,1]
c does not change range of a cos(cx)
-1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
- Definition
- -1 <= sin(anything) <= 1. Multiply by 7: -7 to 7. a + b sin theta: [a-|b|, a+|b|]. Shift then stretch.
- Example
- [-1,1]
- 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 Range of Sine and Cosine.
Explain
Q2. Explain: MAIN IDEA.
Example
Q3. Give one clear example of Range of Sine and Cosine and say why it fits.
Mistake
Q4. State one common mistake students make about Range of Sine and Cosine.
Exam note
Q5. Write a short exam note on Range of Sine and Cosine: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.