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Graphs of Sin and Cos
y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin(x+pi/2)). Amplitude |b| for b sin or b cos. Horizontal stretch 1/|c|. Shift: c(x-h) or cx+d. Meaning: amplitude is the height from the middle; period is the width of one wave.
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Graphs of Sin and Cos
What is it?
y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin(x+pi/2)). Amplitude |b| for b sin or b cos. Horizontal stretch 1/|c|. Shift: c(x-h) or cx+d. Meaning: amplitude is the height from the middle; period is the width of one wave.
In simple words
- y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi.
- y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin(x+pi/2)).
- Amplitude |b| for b sin or b cos.
- The 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
Important terms
GRAPHS OF SIN AND COS: y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin(x+pi/2)). Amplitude |b| for b sin or b cos. Horizontal stretch 1/|c|. Shift: c(x-h) or cx+d. Meaning: amplitude is the height from the middle; period is the width of one wave.
1. PARENT: Five-point sketch: 0, quarter, half, three-quarter, full period. Cos is sine shifted. Both amplitude 1, period 2 pi.
2. a + b sin(cx+d): Middle line y=a. Amplitude |b|. Period 2 pi/|c|. Phase: solve cx+d=0 for a start. Reflect if b<0.
What you should learn
- State the main ideas and notation used in graphs of sin and cos.
- Apply the rules of graphs of sin and cos in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- sin 0=0
- cos 0=1
- Zeros and peaks
PARENT
Five-point sketch: 0, quarter, half, three-quarter, full period. Cos is sine shifted. Both amplitude 1, period 2 pi.
a + b sin(cx+d)
Middle line y=a. Amplitude |b|. Period 2 pi/|c|. Phase: solve cx+d=0 for a start. Reflect if b<0.
Recommended for Class 11
Full school-level explanation
y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin(x+pi/2)). Amplitude |b| for b sin or b cos. Horizontal stretch 1/|c|. Shift: c(x-h) or cx+d. Meaning: amplitude is the height from the middle; period is the width of one wave.
Parent wave, then transforms:
WHAT — GRAPHS OF SIN AND COS
y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin(x+pi/2)). Amplitude |b| for b sin or b cos. Horizontal stretch 1/|c|. Shift: c(x-h) or cx+d. Meaning: amplitude is the height from the middle; period is the width of one wave.
RULE — 1. PARENT
Five-point sketch: 0, quarter, half, three-quarter, full period. Cos is sine shifted. Both amplitude 1, period 2 pi.
METHOD — 2. a + b sin(cx+d)
Middle line y=a. Amplitude |b|. Period 2 pi/|c|. Phase: solve cx+d=0 for a start. Reflect if b<0.
WORKED EXAMPLE
Worked: y = 3 sin(2x). amplitude 3 period 2 pi / 2 = pi middle still y=0 peaks at 3, troughs at -3
PARENT
Five-point sketch: 0, quarter, half, three-quarter, full period. Cos is sine shifted. Both amplitude 1, period 2 pi.
Remember
- five key points
Examples
- sin 0=0
- cos 0=1
- Zeros and peaks
a + b sin(cx+d)
Middle line y=a. Amplitude |b|. Period 2 pi/|c|. Phase: solve cx+d=0 for a start. Reflect if b<0.
Remember
- shift a, stretch |b|
Examples
- a is the axis
- |b| height
- Start when inside=0
How to answer a graphs 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
- amplitude |b|
- period 2 pi/|c| for sin, cos
- amp 3, T=pi
- c is period, b is amplitude
- five key points
- shift a, stretch |b|
- amplitude |b|
- period 2 pi/|c| for sin, cos
How the ideas connect
Worked: y = 3 sin(2x). amplitude 3 period 2 pi / 2 = pi middle still y=0 peaks at 3, troughs at -3
Deeper understanding
Go beyond memorising Graphs of Sin and Cos. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: y = 3 sin(2x). amplitude 3 period 2 pi / 2 = pi middle still y=0 peaks at 3, troughs at -3
Common mistakes
Exam focus
Core Idea
1. PARENT
2. a + b sin(cx+d)
Worked Example
Exam focus
Quick revision
- amplitude |b|
- period 2 pi/|c| for sin, cos
- amp 3, T=pi
- c is period, b is amplitude
- GRAPHS OF SIN AND COS: y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: st
- 1. PARENT: Five-point sketch: 0, quarter, half, three-quarter, full period. Cos is sine shifted. Both amplitude
The 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
For long answers on Graphs of Sin and Cos: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- amplitude |b|
- period 2 pi/|c| for sin, cos
- amp 3, T=pi
- c is period, b is amplitude
Important formulas
- amplitude |b|
- period 2 pi/|c| for sin, cos
- amp 3, T=pi
Common exam traps
- The 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
- 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 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
- 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 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
amplitude |b|
period 2 pi/|c| for sin, cos
amp 3, T=pi
c is period, b is amplitude
y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin
amp 3, T=pi
- Definition
- y=sin x: through origin, up to 1 at pi/2, zero at pi, down to -1 at 3pi/2, back at 2 pi. y=cos x: starts at 1 when x=0, a shift of sin by pi/2 left (cos x = sin
- Formula
- amp 3, T=pi
- Example
- sin 0=0
- Critical warning
- The 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
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Exam Practice
Define
Q1. Define Graphs of Sin and Cos.
Explain
Q2. Explain: PARENT.
Example
Q3. Give one clear example of Graphs of Sin and Cos and say why it fits.
Mistake
Q4. State one common mistake students make about Graphs of Sin and Cos.
Exam note
Q5. Write a short exam note on Graphs of Sin and Cos: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.