AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/3/3-4)
Unit 3 · Topic 3.4
3.4 Sine and Cosine Function Graphs
If you track the y-coordinate of a point going around the unit circle and plot it against the angle, you get the sine graph; tracking the x-coordinate gives the cosine graph. Both are waves with period 2π that swing between −1 and 1.
Key terms
- sine curve
- cosine curve
- period of 2π
- oscillation
- domain and range
Unwrapping the unit circle
The sine function, sin θ, gives the y-coordinate (the vertical displacement from the x-axis) of the point where the terminal ray meets the unit circle. The cosine function, cos θ, gives the x-coordinate (the horizontal displacement from the y-axis). The input θ can be any real number, so both functions have domain all real numbers.
As θ increases and the point travels around the circle, its height rises to 1, falls to −1, and rises again, taking every value in between. So sin θ oscillates between −1 and 1. The same is true for cos θ. Both have range [−1, 1].
After one full turn (2π radians), the point is back where it started, so both graphs repeat every 2π.
Key points
Plot these five points for one cycle, then connect them with a smooth wave:
| θ | 0 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|
| sin θ | 0 | 1 | 0 | −1 | 0 |
| cos θ | 1 | 0 | −1 | 0 | 1 |
Features to know
Sine: zeros at θ = 0, π, 2π, … (every multiple of π). Maximum value 1 at θ = π/2 + 2πk, minimum value −1 at θ = 3π/2 + 2πk, where k is any integer.
Cosine: zeros at θ = π/2, 3π/2, … (π/2 plus any multiple of π). Maximum 1 at θ = 2πk, minimum −1 at θ = π + 2πk.
On [0, 2π], sin θ increases on (0, π/2) and (3π/2, 2π) and decreases on (π/2, 3π/2). It is concave down on (0, π), where it's above the axis, and concave up on (π, 2π).
On [0, 2π], cos θ decreases on (0, π) and increases on (π, 2π). It is concave down on (0, π/2) and (3π/2, 2π), and concave up on (π/2, 3π/2).
The unit circle explains these intervals. sin θ increases exactly when the point is moving up, which happens on the right half of the circle (quadrants IV and I). cos θ increases exactly when the point is moving right, which happens on the bottom half (quadrants III and IV).
Cosine is a shifted sine
The cosine graph is the sine graph shifted left by π/2: cos θ = sin(θ + π/2). For example, sin reaches its peak at π/2 and cos reaches its peak π/2 earlier, at 0.
You can read this on the unit circle too. As the point moves around, its x-coordinate does exactly what its y-coordinate did a quarter turn earlier.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Increasing and concave down
On the interval 0 ≤ θ ≤ 2π, where is sin θ both increasing and concave down?
Show the solutionHide the solution
- Step 1: sin θ increases on (0, π/2) and on (3π/2, 2π).
- Step 2: sin θ is concave down on (0, π), the hump above the axis.
- Step 3: The overlap is (0, π/2). There the height of the point is still rising, but more and more slowly as it nears the top of the circle.
Answer: On 0 < θ < π/2.
- Example 2
Explaining with the unit circle
Use the unit circle to explain why cos θ decreases on 0 < θ < π.
Show the solutionHide the solution
- Step 1: cos θ is the x-coordinate of the point on the unit circle at angle θ.
- Step 2: At θ = 0 the point is (1, 0). As θ increases to π, the point travels counterclockwise over the top of the circle to (−1, 0).
- Step 3: Along the way, the point moves steadily to the left, so its x-coordinate keeps getting smaller, from 1 down to −1.
Answer: The point moves leftward from (1, 0) to (−1, 0) as θ goes from 0 to π, so its x-coordinate, cos θ, decreases.
- Example 3
Trap: bigger input, bigger output?
Without a calculator, decide which is larger: sin 2 or sin 3 (both in radians).
Show the solutionHide the solution
- Step 1: π/2 ≈ 1.571 and π ≈ 3.142, so both 2 and 3 lie between π/2 and π.
- Step 2: On (π/2, π), sin θ is decreasing: the point on the unit circle is coming down from the top.
- Step 3: So the larger input gives the smaller output. The trap is to assume sin 3 is bigger because 3 > 2, or to think in degrees, where sin 2° and sin 3° are both tiny.
Answer: sin 2 > sin 3 (in fact sin 2 ≈ 0.909 and sin 3 ≈ 0.141).
Common mistakes
- Assuming the input is in degrees. On the graph, θ = 3 means 3 radians, a little less than π.
- Giving sine's zeros as π/2 + kπ. Those are cosine's zeros; sine's are at multiples of π.
- Shifting the wrong way: cos θ = sin(θ + π/2) is sine shifted left, not right.
On the exam
- Expect questions on where sin θ or cos θ is increasing, decreasing, or concave up or down, often explained with the unit circle.
- Questions may compare values like sin 2 and sin 3 without a calculator. Locate each input relative to π/2, π and 3π/2.
Connected topics
Videos
Check yourself
4 questions on 3.4 Sine and Cosine Function Graphs. Pick an answer to see if you got it, and why.
On which of the following intervals are both y = sin θ and y = cos θ decreasing?
Which of the following is equal to cos θ for all θ?
On which of the following intervals is y = sin θ both increasing and concave down?
At which of the following values of θ do the graphs of y = sin θ and y = cos θ intersect?
0 of 4 answered