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3.4 · sin θ Graph
Where does f(θ)=sin θ start, peak, cross zero, and reach minimum?
Think: unit circle y-coordinates at 0, π/2, π, 3π/2, 2π
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3.4 · cos θ Graph
Where does g(θ)=cos θ start, peak, cross zero, and reach minimum?
Think: unit circle x-coordinates at 0, π/2, π, 3π/2, 2π
3/10
3.4 · sin vs cos
How does cos θ relate to sin θ? Write the identity.
Phase shift left or right?
4/10
3.4 · Midline Formula
How do you find the midline of a sinusoidal function from its max and min?
y = ?
Average of max and min
5/10
3.4 · Amplitude Formula
How do you find the amplitude from max and min?
Distance from midline to max
6/10
3.4 · Period vs Frequency
How are period P and frequency f related? What does each measure?
They are reciprocals
7/10
3.4 · Max to Min
A sinusoidal function has max at θ=π and min at θ=3π. What is the period?
Distance max→min = 2π
Max to min is HALF the period
8/10
3.4 · Example 3
h(θ): max at (π,8), min at (3π,−2). State ALL FOUR properties.
Period, frequency, midline, amplitude
2/10
✓ cos θ Key Points
Starts at MAX (0,1) · zero at (π/2,0) · min at (π,−1) · zero at (3π/2,0) · back to max at (2π,1)
cos(0)=1 (starts at top). Drops to zero at π/2. Reaches minimum −1 at π. Returns to 1 at 2π.
1/10
✓ sin θ Key Points
Starts at ZERO (0,0) · max at (π/2,1) · zero at (π,0) · min at (3π/2,−1) · zero at (2π,0)
sin(0)=0 (starts on midline). Peaks at π/2. Returns to zero at π. Minimum at 3π/2.
4/10
✓ Midline Formula
Midline = (max + min) / 2
The midline is the AVERAGE of the maximum and minimum values. It is the horizontal axis of symmetry of the wave.
3/10
✓ sin vs cos Identity
cos θ = sin(θ + π/2) — cosine is sine shifted LEFT by π/2
Both have same period (2π), amplitude (1), and midline (y=0). Cosine is just ahead of sine by a quarter-period.
6/10
✓ Example 3 — h(θ)
Period=4π · Frequency=1/(4π) · Midline y=3 · Amplitude=5
Half-period=3π−π=2π so period=4π. Midline=(8+(−2))/2=3. Amplitude=8−3=5. (Example 3.)
5/10
✓ Amplitude Formula
Amplitude = max − midline = (max − min) / 2
Either subtract midline from max, OR divide the total range (max−min) by 2. Both give the same answer.
8/10
✓ Sinusoidal Function
Any additive/multiplicative transformation of f(θ)=sin θ. Cosine is sinusoidal: cos θ=sin(θ+π/2)
A sinusoidal function can always be written in terms of sine. Cosine looks different but IS a sine function shifted left by π/2.
7/10
✓ Period vs Frequency
Period P = length of one full cycle · Frequency = 1/P (cycles per unit)
Larger period = slower oscillation = smaller frequency. They are reciprocals: P × f = 1.
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3.4 · Sinusoidal Def
What is a sinusoidal function? Give an example of one that is NOT obviously sine.
Think about cosine
10/10
3.4 · Clock Example
Clock center=120in, hand=8in, period=30min. What is the amplitude, midline, and the 5 key points?
Max=center+hand, min=center−hand
10/10
✓ Clock Key Points
F(0,128) G(7.5,120) J(15,112) K(22.5,120) P(30,128) Amplitude=8 Midline y=120
Max=120+8=128 (hand up). Min=120−8=112 (hand down). Quarter-period=7.5 min. Midline=120 (center height).
9/10
✓ Max to Min = Half
Period = 2 × (distance from max to min) = 2 × 2π = 4π
Max to the NEXT min spans only HALF a period. Always double it! Distance=3π−π=2π → period=4π.