Source: AP Precalc 3.4 Notes (The Algebros / FlippedMath)

Tags: sine function, sin θ, unit circle, y-coordinate, periodic function, sine graph, sine table of values, trig graphs, AP Precalculus 3.4

TL;DR

The sine function f(θ) = sin θ gives the y-coordinate of a point on the unit circle for a given angle θ. Its graph forms a smooth, repeating wave between −1 and 1 with a period of 2π. Understanding where sine increases, decreases, and changes concavity is essential for describing its behaviour on any interval.

Key Terms

Sine function / sin θ

For an angle θ in standard position on the unit circle, sin θ equals the y-coordinate of the point where the terminal ray meets the circle. Equivalently, it is the vertical displacement from the x-axis.

Unit circle

A circle of radius 1 centred at the origin. Every point on it can be written as (cos θ, sin θ).

Standard position (of an angle)

An angle whose vertex is at the origin and whose initial ray lies along the positive x-axis. The terminal ray rotates anticlockwise for positive angles.

Periodic function

A function that repeats its values at regular intervals. For sin θ, the period is 2π, meaning sin(θ + 2π) = sin θ for all θ.

Period

The horizontal length of one complete cycle of a periodic function. For sin θ, one full cycle runs from 0 to 2π.

Concavity

Describes the curvature of a graph. Concave up looks like a cup (curves upward); concave down looks like a cap (curves downward).

Core Content

Unit Circle Coordinates and Sine Values

Each point on the unit circle is written (cos θ, sin θ). The sine value is simply the y-coordinate.

• At θ = 0: the point is (1, 0), so sin 0 = 0

• At θ = π/2: the point is (0, 1), so sin(π/2) = 1

• At θ = π: the point is (−1, 0), so sin π = 0

• At θ = 3π/2: the point is (0, −1), so sin(3π/2) = −1

• At θ = 2π: back to (1, 0), so sin 2π = 0

Sine Table of Values (0 to 2π)

First half (0 to π):

Second half (π to 2π):

Notice the symmetry: the second half of the cycle mirrors the first half but with negative values. The wave rises from 0 to 1, falls back through 0 to −1, then returns to 0.

Shape of the Sine Graph

Plotting these points produces the classic sine wave:

• Starts at the origin (0, 0)

• Rises to a maximum of 1 at θ = π/2

• Returns to 0 at θ = π

• Falls to a minimum of −1 at θ = 3π/2

• Returns to 0 at θ = 2π

This completes one cycle. The pattern then repeats indefinitely in both directions.

Increasing and Decreasing Intervals for sin θ

sin θ is increasing on:

• 0 ≤ θ ≤ π/2 (first quarter of the cycle, rising from 0 to 1)

• 3π/2 ≤ θ ≤ 2π (last quarter, rising from −1 back to 0)

sin θ is decreasing on:

• π/2 ≤ θ ≤ 3π/2 (the middle half, falling from 1 down to −1)

A useful way to remember: sine increases when the unit circle point is moving upward (Quadrants I and IV) and decreases when it moves downward (Quadrants II and III).

Concavity of sin θ

Concave down on 0 ≤ θ ≤ π

The graph curves like an upside-down bowl through its peak at π/2.

Concave up on π ≤ θ ≤ 2π

The graph curves like a bowl through its trough at 3π/2.

The concavity changes at θ = π (and at θ = 0 / 2π), which are the points where sin θ = 0. These are inflection points of the sine curve.

Combining Increasing/Decreasing with Concavity

This is the type of description AP Precalculus expects you to produce for any sub-interval:

• 0 < θ < π/2: increasing and concave down

• π/2 < θ < π: decreasing and concave down

• π < θ < 3π/2: decreasing and concave up

• 3π/2 < θ < 2π: increasing and concave up

For a wider interval like π < θ < 2π, you combine: concave up throughout, decreasing from π to 3π/2, then increasing from 3π/2 to 2π.

Formulas / Diagrams

Key formula:

f(θ) = sin θ = y-coordinate on the unit circle

Reference values worth memorising:

• sin 0 = 0

• sin(π/6) = 1/2

• sin(π/4) = √2/2 ≈ 0.707

• sin(π/3) = √3/2 ≈ 0.866

• sin(π/2) = 1

All other values in the 0 to 2π range follow from symmetry and sign changes by quadrant.

Why It Matters / Exam Flags

⚠️ The AP exam frequently asks you to describe the behaviour of sin θ (or cos θ) on a specific interval, combining both increasing/decreasing and concavity in your answer.

⚠️ Remember that sin θ gives the y-coordinate, not the x-coordinate. Mixing this up is an easy mark to lose.

⚠️ The sine function's concavity flips at θ = π (and multiples of π). The increasing/decreasing behaviour flips at θ = π/2 and θ = 3π/2. These are different transition points, so do not conflate them.

⚠️ When a question shows an angle θ on the unit circle diagram and asks about another angle α in a given range, you need to determine whether the function is monotonically increasing or decreasing across that entire range, or whether it changes direction within it.

Practice Q&A

Q: On what interval(s) is f(θ) = sin θ increasing on [0, 2π]?

A: 0 ≤ θ ≤ π/2 and 3π/2 ≤ θ ≤ 2π.

Q: Describe the concavity and increasing/decreasing behaviour of sin θ on the interval π/2 < θ < π.

A: On this interval, sin θ is decreasing and concave down.

Q: Describe the concavity and increasing/decreasing behaviour of sin θ on the interval 3π/2 < θ < 2π.

A: On this interval, sin θ is increasing and concave up.

Q: Describe the concavity and increasing/decreasing behaviour of sin θ on the interval π/2 < θ < 3π/2.

A: Concave down from π/2 to π, then concave up from π to 3π/2. Decreasing throughout the entire interval.

Q: What is sin θ geometrically on the unit circle?

A: The y-coordinate of the point where the terminal ray of angle θ intersects the unit circle (i.e. the vertical displacement from the x-axis).

Q: For the function g(a) = sin a, if θ is in Quadrant III (between π and 3π/2) and θ < α < 3π/2, is g(α) < g(θ), g(α) > g(θ), or does it depend?

A: g(α) < g(θ). On the interval from π to 3π/2, sine is decreasing, so a larger angle gives a smaller (more negative) sine value.

Related Terms / Search Tags

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