1/10
3.4 · sin θ Key Points
State the 5 key points of f(θ) = sin θ over one period.
Start at (0,?), peak at (π/2,?), zero at (π,?), min at (3π/2,?), back at (2π,?)
2/10
3.4 · cos θ Key Points
State the 5 key points of g(θ) = cos θ over one period.
Start at (0,?), zero at (π/2,?), min at (π,?), zero at (3π/2,?), back at (2π,?)
3/10
3.4 · sin vs cos Start
Where does sin θ start? Where does cos θ start? Why is this the key difference?
4/10
3.4 · sin vs cos Identity
Write cos θ as a transformation of sin θ. What kind of transformation is it?
cos θ = sin( ? )
5/10
3.4 · Properties of sin θ
State the midline, amplitude, period, and frequency of f(θ) = sin θ.
6/10
3.4 · Properties of cos θ
State the midline, amplitude, period, and frequency of g(θ) = cos θ.
7/10
3.4 · Concavity of sin θ
On which interval in [0, 2π] is f(θ) = sin θ concave DOWN?
Concave down = hill shape = where the graph curves downward
8/10
3.4 · Concavity of cos θ
On which interval in [0, 2π] is g(θ) = cos θ concave UP?
Concave up = bowl shape = where the graph curves upward
2/10
✓ cos θ Key Points
(0,1) → (π/2,0) → (π,−1) → (3π/2,0) → (2π,1)
cos starts at MAX (1). Drops to zero at π/2. Minimum −1 at π. Returns to zero at 3π/2. Back to 1 at 2π.
1/10
✓ sin θ Key Points
(0,0) → (π/2,1) → (π,0) → (3π/2,−1) → (2π,0)
sin starts at ZERO (midline). Peaks to 1 at π/2. Returns to zero at π. Minimum −1 at 3π/2. Back to zero at 2π.
4/10
✓ sin vs cos Identity
cos θ = sin(θ + π/2) — horizontal shift LEFT by π/2
Cosine is sine shifted left by a quarter-period. Both have the same amplitude, period, and midline.
3/10
✓ sin vs cos Start
sin θ starts at ZERO (midline). cos θ starts at MAXIMUM (1).
This is the key difference: sin(0)=0, cos(0)=1. Think: Sin starts at the midline, Cos starts at the Crest.
6/10
✓ Properties of cos θ
Midline y=0 · Amplitude=1 · Period=2π · Frequency=1/(2π)
Identical to sin θ in all four properties. The only difference is the starting point (phase).
5/10
✓ Properties of sin θ
Midline y=0 · Amplitude=1 · Period=2π · Frequency=1/(2π)
Midline=(1+(−1))/2=0. Amplitude=1−0=1. One full cycle spans 0 to 2π so period=2π. Frequency=1/(2π).
8/10
✓ Concavity of cos θ
Concave UP on (π/2, 3π/2)
The lower half of the cos wave (from the descending zero-crossing to the ascending zero-crossing) is bowl-shaped = concave up. This is the interval containing the minimum at π.
7/10
✓ Concavity of sin θ
Concave DOWN on (0, π)
The upper half of the sin wave (from start to π) is hill-shaped = concave down. The lower half (π to 2π) is bowl-shaped = concave up.
9/10
3.4 · Increasing sin θ
On which intervals in [0, 2π] is f(θ) = sin θ increasing?
sin is increasing when it goes uphill
10/10
3.4 · Decreasing cos θ
On which interval in [0, 2π] is g(θ) = cos θ decreasing?
cos decreases from its peak toward its minimum
10/10
✓ Decreasing cos θ
Decreasing on (0, π)
cos starts at its maximum (1) at θ=0, decreases to its minimum (−1) at θ=π, then increases back to 1 at 2π. So it's decreasing on the interval (0, π).
9/10
✓ Increasing sin θ
Increasing on (−π/2, π/2) within one period, or equivalently (0, π/2) and (3π/2, 2π) over [0, 2π]
sin increases as it goes from minimum (3π/2) through zero (0) up to maximum (π/2). Then decreases from π/2 to 3π/2.