MathematicsChapter 9: Trigonometric Functions
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.
Learning objectives
- 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.
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(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.
Mathematical form
y=sin x
y=cos x
x = sin(x+pi/2)). Amplitude
y = 3 sin(2x). amplitude 3 period 2 pi / 2
Where / symbols
- 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.
Overview
Parent wave, then transforms:
Parent
Five-point sketch: 0, quarter, half, three-quarter, full period. Cos is sine shifted. Both amplitude 1, period 2 pi.
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.
Examples a is the axis · |b| height · Start when inside=0
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
How to answer a graphs 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
- y = 3 sin(2x).
- amplitude 3
- period 2 pi / 2 = pi
- middle still y=0
- peaks at 3, troughs at -3
In simple words
The 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
Key points
- amplitude |b|
- period 2 pi/|c| for sin, cos
- five key points
- shift a, stretch |b|
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Graphs of Sin and Cos
Sticky Notes · Quick Revision
Core Idea
- 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.
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.
Worked Example
- Worked: y = 3 sin(2x).
- amplitude 3
- period 2 pi / 2 = pi
- middle still y=0
- peaks at 3, troughs at -3
Common Misconception
- The 2 in sin(2x) squeezes the period to pi. It does not make the amplitude 2.
FBISE Exam Check
- State the rule for graphs of sin and cos before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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