3.4 Sine and Cosine Function Graphs

The sine and cosine functions come straight from the unit circle: for an angle θ in standard position, sin θ is the y-coordinate of the point where the terminal ray meets the circle, and cos θ is the x-coordinate. As θ increases, both outputs oscillate between -1 and 1, and tracing those coordinates around the circle gives you the smooth, repeating wave graphs of y = sin θ and y = cos θ.

Why This Matters for the AP Precalculus Exam

Trigonometric and polar functions make up the largest share of the AP Precalculus exam, and this topic is the foundation for everything sinusoidal that follows. Once you can connect a point moving around the unit circle to a height (sine) or a horizontal position (cosine), you can read and build sine and cosine graphs, identify their key features, and later handle amplitude, period, phase shift, and real-world models.

On the exam you may be asked to construct or recognize representations of sine and cosine, match a unit-circle position to a point on the graph, or explain why an output value makes sense for a given angle. These skills show up in both calculator and non-calculator settings, so building fluency here pays off across the whole unit.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculator

Key Takeaways

• For an angle θ in standard position, the terminal ray hits the unit circle at point P, where sin θ is the y-coordinate and cos θ is the x-coordinate of P.
• The domain of both sine and cosine is all real numbers, and the outputs oscillate between -1 and 1, hitting every value in between.
• sin θ tracks the vertical distance of P from the x-axis; cos θ tracks the horizontal distance of P from the y-axis.
• y = sin θ starts at (0, 0), reaches a maximum at θ = π/2, returns to 0 at θ = π, hits a minimum at θ = 3π/2, and returns to 0 at θ = 2π.
• y = cos θ starts at (0, 1), is 0 at θ = π/2, reaches a minimum at θ = π, is 0 at θ = 3π/2, and returns to 1 at θ = 2π.
• Where one function is 0, the other is at a maximum or minimum, which is why their graphs look like shifted copies of each other.

Patterns in the Unit Circle

Recall the unit circle: a circle with radius 1 centered at the origin. For an angle θ in standard position, the terminal ray meets the circle at a point P. The coordinates of P are exactly $(\cos θ, \sin θ)$, so the whole graph of each function comes from watching one coordinate of P change as you rotate counterclockwise.

The cosine function gives the x-coordinate of P. As θ goes from 0 to π radians, the cosine values decrease, because the horizontal position of P moves from the far right toward the far left. After θ = π/2 the values become negative, reaching their most negative at θ = π. From π to 2π, the cosine values increase again as P moves back to the right through the third and fourth quadrants.

One thing to keep straight: the magnitude (absolute value) of the horizontal distance is not always increasing or decreasing, but the signed x-coordinate is what the cosine function reports. Values can go from negative to less negative (increasing) or negative to more negative (decreasing).

Cosine values range from 1 to -1. The maximum of 1 happens at the point (1, 0), which corresponds to θ = 0, and the minimum of -1 happens at (-1, 0), which corresponds to θ = π. At θ = π/2 and θ = 3π/2, the cosine value is 0.

The sine function gives the y-coordinate of P. As θ goes from 0 to π/2 radians, the sine values increase, because the vertical position of P rises through the first quadrant. After π/2 the sine values decrease until θ = 3π/2, since the vertical position falls through the second and third quadrants. From 3π/2 to 2π, the sine values increase again as P climbs back toward the x-axis.

Sine values also range from 1 to -1. The maximum of 1 happens at (0, 1), which corresponds to θ = π/2, and the minimum of -1 happens at (0, -1), which corresponds to θ = 3π/2. At θ = 0 and θ = π, the sine value is 0.

Compare the key points of sine and cosine and you'll see they're offset: when cosine is 0, sine is at a maximum or minimum, and when cosine is at a maximum or minimum, sine is 0. Learning the behavior at these key angles for both functions makes graphing much faster.

Constructing the Sine Curve

The sine function takes an angle θ as input and returns the y-coordinate of point P on the unit circle, a value between -1 and 1. In function notation, you write it as $f(θ) = \sin θ$.

To build the sine curve, plot the angle θ on the horizontal axis (independent variable) and the sine value on the vertical axis (dependent variable). Use the unit-circle patterns to find the outputs:

• At θ = 0, sin θ = 0, so plot the origin (0, 0).
• At θ = π/2, sin θ = 1, so plot the maximum point $(\frac{π}{2}, 1)$.
• At θ = π, sin θ = 0, so $(π, 0)$ is an x-intercept.
• At θ = 3π/2, sin θ = -1, so plot the minimum point $(\frac{3π}{2}, -1)$.
• At θ = 2π, sin θ = 0, so $(2π, 0)$ is another x-intercept.

Connect these points with a smooth curve and you get the wave that oscillates between -1 and 1. Because the domain is all real numbers, the curve keeps going in both directions, repeating the same shape every $2π$ radians (the function is periodic).

Constructing the Cosine Curve

The cosine function takes an angle θ as input and returns the x-coordinate of point P on the unit circle, again a value between -1 and 1. In function notation, $f(θ) = \cos θ$.

To build the cosine curve, plot θ on the horizontal axis and the cosine value on the vertical axis. Use the same unit-circle patterns:

• At θ = 0, cos θ = 1, so plot $(0, 1)$.
• At θ = π/2, cos θ = 0, so $(\frac{π}{2}, 0)$ is an x-intercept.
• At θ = π, cos θ = -1, so plot the minimum point $(π, -1)$.
• At θ = 3π/2, cos θ = 0, so $(\frac{3π}{2}, 0)$ is another x-intercept.
• At θ = 2π, cos θ = 1, so the curve returns to $(2π, 1)$.

Connect these points with a smooth curve. Like sine, cosine oscillates between -1 and 1, has all real numbers as its domain, and is periodic, repeating every $2π$ radians. The cosine graph looks like the sine graph shifted to the left, which matches the fact that the two functions reach their key values at offset angles.

How to Use This on the AP Precalculus Exam

Problem Solving

• Start from the point P. If you can place the terminal ray on the unit circle, read sin θ off the y-axis and cos θ off the x-axis. This works even for angles you don't have memorized.
• Memorize the five key points for one full period of each function. They let you sketch a clean graph or check a multiple-choice graph fast.
• When asked to construct a representation, label the axes clearly: θ across the bottom, output value up the side. Clear setup makes your reasoning easy to follow.

Common Trap

• Mixing up which coordinate goes with which function. Cosine = x-coordinate, sine = y-coordinate. Keep that straight before you do anything else.
• Forgetting that the outputs are capped between -1 and 1. Any answer outside that range for plain sine or cosine is a signal to recheck.

Common Misconceptions

• Sine and cosine are not just about right triangles. Right-triangle ratios are where the idea starts, but on the unit circle sine and cosine are defined for any angle, including negative angles and angles past 2π. Their domain is all real numbers.
• The maximum of sine and cosine is 1, not π/2. Students sometimes confuse the input that produces the maximum with the maximum itself. For y = sin θ, the maximum output is 1 and it occurs at θ = π/2.
• Increasing cosine doesn't mean the point is moving away from the origin. On the unit circle every point is distance 1 from the origin. Cosine increasing means the signed x-coordinate is getting larger, even when that just means going from a more negative to a less negative value.
• Sine and cosine are different functions, not the same graph relabeled. They share the same shape and range, but their key points are offset. At a given θ, one can be 0 while the other is at a peak.
• The period is 2π, not π. A full trip around the unit circle is 2π radians, so the sine and cosine graphs repeat every 2π, not every π.

Related AP Precalculus Guides

• 3.3 Sine and Cosine Function Values
• 3.6 Sinusoidal Function Transformations
• 3.9 Inverse Trigonometric Functions
• 3.8 The Tangent Function
• 3.11 The Secant, Cosecant, and Cotangent Functions
• 3.12 Equivalent Representations of Trigonometric Functions