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

Tags: cosine function, cos θ, unit circle, x-coordinate, periodic function, cosine graph, cosine table of values, trig graphs, AP Precalculus 3.4

TL;DR

The cosine function f(θ) = cos θ gives the x-coordinate of a point on the unit circle for a given angle θ. Its graph is the same wave shape as sine but shifted left by π/2. Cosine starts at its maximum (1), decreases to −1, then returns to 1 over one full period of 2π. Knowing its increasing/decreasing intervals and concavity is tested the same way as sine on the AP exam.

Key Terms

Cosine function / cos θ

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

Horizontal displacement

The signed distance of the unit circle point from the y-axis. Positive to the right, negative to the left.

Phase shift (sine vs cosine relationship)

Cosine is a shifted version of sine: cos θ = sin(θ + π/2). The graphs are identical in shape but offset horizontally. This is worth knowing but the notes focus on each function independently.

Core Content

Unit Circle Coordinates and Cosine Values

Each point on the unit circle is (cos θ, sin θ). The cosine value is the x-coordinate.

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

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

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

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

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

Cosine Table of Values (0 to 2π)

First half (0 to π):

Second half (π to 2π):

The cosine values in the second half mirror the first half but in reverse order, moving from −1 back up to 1.

Shape of the Cosine Graph

• Starts at the maximum: (0, 1)

• Falls to 0 at θ = π/2

• Reaches its minimum of −1 at θ = π

• Returns to 0 at θ = 3π/2

• Rises back to the maximum of 1 at θ = 2π

Compare with sine: sine starts at 0 and peaks at π/2, while cosine starts at 1 and hits zero at π/2. Same wave, different starting point.

Increasing and Decreasing Intervals for cos θ

cos θ is decreasing on:

• 0 ≤ θ ≤ π (the entire first half of the cycle, falling from 1 to −1)

cos θ is increasing on:

• π ≤ θ ≤ 2π (the entire second half, rising from −1 back to 1)

This is simpler to remember than sine: cosine decreases across the first half-period and increases across the second.

Concavity of cos θ

Concave up on π/2 ≤ θ ≤ 3π/2

The graph curves like a bowl through its minimum at π.

Concave down on 0 ≤ θ ≤ π/2 and 3π/2 ≤ θ ≤ 2π

The graph curves like a cap near its maximum values.

The concavity changes at θ = π/2 and θ = 3π/2, where cos θ = 0. These are inflection points of the cosine curve.

Combining Increasing/Decreasing with Concavity

The four quadrant-by-quadrant descriptions for cos θ:

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

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

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

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

For a wider interval like 0 < θ < π: decreasing throughout, concave down from 0 to π/2 then concave up from π/2 to π.

Formulas / Diagrams

Key formula:

f(θ) = cos θ = x-coordinate on the unit circle

Reference values worth memorising:

• cos 0 = 1

• cos(π/6) = √3/2 ≈ 0.866

• cos(π/4) = √2/2 ≈ 0.707

• cos(π/3) = 1/2

• cos(π/2) = 0

Sine vs Cosine Quick Comparison

Why It Matters / Exam Flags

⚠️ Cosine gives the x-coordinate, sine gives the y-coordinate. This is tested directly and indirectly throughout AP Precalculus.

⚠️ When asked about cos θ on a specific interval, check both increasing/decreasing and concavity separately. They have different transition points (π for inc/dec, π/2 and 3π/2 for concavity).

⚠️ Questions showing an angle θ on a unit circle diagram and asking about g(α) = cos a for α in some range require you to determine whether cosine is monotonically changing across that range:

• If cosine is strictly increasing or strictly decreasing on the interval from θ to the boundary, you can compare g(α) and g(θ) directly.

• If the interval spans a transition point (e.g. it crosses π), the answer may be "depends on the value of α."

⚠️ The domain values where cos θ = 1 are θ = 2πn (every full revolution). The domain values where sin θ = 0 are θ = nπ (every half revolution). These are common test prep questions.

Practice Q&A

Q: On what interval is f(θ) = cos θ decreasing on [0, 2π]?

A: 0 ≤ θ ≤ π.

Q: Describe the concavity and increasing/decreasing behaviour of cos θ on the interval 0 < θ < π/2.

A: Decreasing and concave down.

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

A: Decreasing and concave up.

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

A: Increasing and concave up.

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

A: Increasing and concave down.

Q: Describe the behaviour of cos θ on the full interval 0 < θ < π.

A: Decreasing throughout. Concave down from 0 to π/2, then concave up from π/2 to π.

Q: For f(θ) = cos θ, what are all values of the domain where f(θ) = 1?

A: θ = 2nπ, where n is any integer. That is: ..., −2π, 0, 2π, 4π, ...

Q: For g(θ) = sin θ, what are all values of the domain where g(θ) = 0?

A: θ = nπ, where n is any integer. That is: ..., −π, 0, π, 2π, ...

Q: We are given angle θ in Quadrant I (between 0 and π/2) on the unit circle. For g(a) = cos a and α where θ < α < 3π/2, is g(α) < g(θ)?

A: Yes. Cosine is decreasing on [0, π], so any angle α > θ that is still within [0, π] will have a smaller cosine value. Beyond π, cosine is negative and increasing but still below cos θ (which is positive in Q1). By 3π/2, cos(3π/2) = 0, which is still less than a positive cos θ. So g(α) < g(θ) throughout.

Q: What is the geometric meaning of cos θ on the unit circle?

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

Related Terms / Search Tags

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