Sine and cosine function values (unit circle) — AP Precalculus Study Guide

For: AP Precalculus candidates sitting AP Precalculus.

Covers: Unit circle definition of sine and cosine, reference angle calculation, quadrant sign rules, finding exact values for common angles, coterminal angles, and using the Pythagorean identity to find unknown trigonometric values.

You should already know: Radians and degree angle conversion, coordinate plane point plotting, basic right triangle trigonometry.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Precalculus style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Sine and cosine function values (unit circle)?

The unit circle definition of sine and cosine extends right-triangle trigonometry from acute angles to all real-number angles, making it possible to model periodic phenomena and work with trigonometric functions of any angle. Per the AP Precalculus CED, this topic is core content in Unit 3, making up approximately 2-4% of the total exam score, with questions appearing in both multiple-choice (MCQ) and free-response (FRQ) sections. By definition, for any angle θ (measured in radians counterclockwise from the positive x-axis), the terminal side of θ intersects the unit circle (centered at the origin with radius 1) at the point $P(x,y)$. By convention, $\cos \theta =x$ and $\sin \theta =y$. This definition is sometimes called the circular definition of sine and cosine, and it is the foundation for nearly all other trigonometric concepts you will encounter on the exam.

2. Unit Circle Definition of Sine and Cosine

The unit circle is the set of all points $(x,y)$ that satisfy the equation: $$x^2+y^2=1$$ When working with angles on the unit circle, we follow standard position convention: an angle is measured starting from the positive x-axis, with counterclockwise rotation representing positive angles and clockwise rotation representing negative angles. The intersection of the angle's terminal side (the line marking the end of the rotation) with the unit circle gives the values of sine and cosine directly: $\cos \theta$ equals the x-coordinate of the intersection point, and $\sin \theta$ equals the y-coordinate. This definition matches the right-triangle definition for acute angles: for an acute angle in a right triangle with hypotenuse 1, the adjacent side (to the angle) is the x-coordinate, so $\cos \theta =\text{adjacent}/\text{hypotenuse}=x$, and the opposite side is the y-coordinate, so $\sin \theta =\text{opposite}/\text{hypotenuse}=y$. The definition extends to all real angles, however, including negative angles and angles greater than $90^{\circ }$ ($\pi /2$ radians).

Worked Example

State the coordinates of the intersection of the terminal side of $\theta =7\pi /6$ with the unit circle, then give the values of $\cos \left(\frac{7\pi }{6}\right)$ and $\sin \left(\frac{7\pi }{6}\right)$.

1. First, identify the quadrant: $7\pi /6$ is between $\pi$ and $3\pi /2$, so it lies in Quadrant III. In Quadrant III, both x and y coordinates are negative.
2. The reference angle (acute angle between the terminal side and the x-axis) for $7\pi /6$ is $7\pi /6-\pi =\pi /6$.
3. For $\pi /6$ in Quadrant I, we know the intersection point is $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$.
4. Apply the Quadrant III sign rule: both coordinates are negative, so the intersection point is $\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$.
5. By definition, $\cos \left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}$ and $\sin \left(\frac{7\pi }{6}\right)=-\frac{1}{2}$.

Exam tip: On MCQs, you can often eliminate two wrong options immediately just by checking the sign of sine and cosine based on quadrant, before doing any calculation to find the magnitude.

3. Reference Angles and Exact Values for Any Angle

A reference angle is the acute angle that the terminal side of any angle makes with the x-axis, and it always has a measure between $0$ and $\pi /2$. Due to the symmetry of the unit circle, the absolute value of sine and cosine for any angle is equal to the sine and cosine of its reference angle. Only the sign of the value changes, based on which quadrant the angle falls into.

To find the exact value of sine or cosine for any angle:

1. If the angle is negative or larger than $2\pi$, find a coterminal angle between $0$ and $2\pi$ by adding or subtracting integer multiples of $2\pi$.
2. Identify the quadrant of the coterminal angle, to get the correct sign of the final value.
3. Calculate the reference angle $\alpha$ using quadrant-specific rules: Q1: $\alpha =\theta$; Q2: $\alpha =\pi -\theta$; Q3: $\alpha =\theta -\pi$; Q4: $\alpha =2\pi -\theta$.
4. Use the known value of sine/cosine for $\alpha$, and apply the correct sign from step 2.

Worked Example

Find the exact value of $\sin \left(-\frac{7\pi }{4}\right)$.

1. Find a positive coterminal angle between $0$ and $2\pi$: add $2\pi =8\pi /4$ to the negative angle: $-\frac{7\pi }{4}+\frac{8\pi }{4}=\frac{\pi }{4}$.
2. $\pi /4$ is in Quadrant I, where sine (y-coordinate) is positive.
3. The reference angle for a Q1 angle is the angle itself, so $\alpha =\pi /4$.
4. We know $\sin (\pi /4)=\frac{\sqrt{2}}{2}$, so the final value is $\sin \left(-\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2}$.

Exam tip: Always reduce radian fractions to their simplest form immediately. For example, rewrite $10\pi /8$ as $5\pi /4$ right away, to avoid miscounting quadrants or misidentifying common angles.

4. Find Unknown Sine/Cosine Values with the Pythagorean Identity

The Pythagorean identity for sine and cosine is derived directly from the unit circle equation. Since $x^2+y^2=1$, and $x=\cos \theta$, $y=\sin \theta$, we get: $${\cos }^2\theta +{\sin }^2\theta =1$$ This identity holds for all real angles $\theta$, and it is a common exam tool to find an unknown sine or cosine value when you know the other value and the quadrant of $\theta$. The key step after solving for the squared value is to pick the correct sign based on the quadrant of $\theta$, since taking the square root gives both a positive and negative solution.

Worked Example

Given that $\cos \theta =\frac{2}{3}$ and $\theta$ is in Quadrant IV, find $\sin \theta$.

1. Substitute the known value of $\cos \theta$ into the Pythagorean identity: $${\left(\frac{2}{3}\right)}^2+{\sin }^2\theta =1$$
2. Simplify and solve for ${\sin }^2\theta$: $$\frac{4}{9}+{\sin }^2\theta =1⟹{\sin }^2\theta =1-\frac{4}{9}=\frac{5}{9}$$
3. Take the square root of both sides to get two possible solutions: $$\sin \theta =\pm \frac{\sqrt{5}}{3}$$
4. $\theta$ is in Quadrant IV, where y-coordinates (and thus $\sin \theta$) are negative, so the final solution is $\sin \theta =-\frac{\sqrt{5}}{3}$.

Exam tip: Never skip writing the $\pm$ when taking the square root. Explicitly writing the sign option reminds you to select the correct sign based on quadrant, which is the most commonly missed point on this problem type.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Calculates the correct magnitude of sine/cosine from the reference angle, then leaves the value positive regardless of quadrant. For example, leaving $\cos (5\pi /4)=\sqrt{2}/2$ instead of $-\sqrt{2}/2$. Why: Students stop after recalling the common angle value and forget to apply the sign rule. Correct move: Always after finding the magnitude, explicitly state the quadrant and assign the correct sign before writing your final answer.
• Wrong move: Misremembers the reference angle formula for Quadrant IV, using the Quadrant III rule. For example, finding the reference angle for $5\pi /3$ as $5\pi /3-\pi =2\pi /3$ instead of $2\pi -5\pi /3=\pi /3$. Why: Students mix up the order of subtraction for Q3 vs Q4. Correct move: For any coterminal angle in $[0,2\pi )$, first write down which quadrant it is in, then use the matching reference angle formula for that quadrant.
• Wrong move: When solving for $\sin \theta$, uses the sign of $\cos \theta$ to pick the sign of $\sin \theta$, instead of the quadrant. For example, if $\cos \theta$ is positive, incorrectly assuming $\sin \theta$ must also be positive. Why: Students forget that sine and cosine have opposite signs in Q2 and Q4. Correct move: Always assign the sign of the unknown function based solely on the given quadrant of $\theta$, not the sign of the known function.
• Wrong move: Tries to calculate a reference angle for an angle larger than $2\pi$ or negative without first finding a coterminal angle between $0$ and $2\pi$. Why: Students think they can subtract $\pi$ directly from a large angle, leading to an incorrect reference angle. Correct move: For any angle outside $[0,2\pi )$, first add or subtract multiples of $2\pi$ to get a coterminal angle in the correct range before calculating the reference angle.
• Wrong move: Swaps sine and cosine, writing $\sin \theta =x$ and $\cos \theta =y$ for the unit circle point $(x,y)$. Why: Students misremember the order when memorizing the definition. Correct move: Use the mnemonic "cos(x), sin(y)" to always recall that cosine maps to the x-coordinate and sine maps to the y-coordinate.

6. Practice Questions (AP Precalculus Style)

Question 1 (Multiple Choice)

Which of the following is equal to $\sin \left(\frac{17\pi }{4}\right)$? A) $\frac{\sqrt{2}}{2}$ B) $-\frac{\sqrt{2}}{2}$ C) $\frac{1}{2}$ D) $-\frac{1}{2}$

Worked Solution: First, find a coterminal angle of $\frac{17\pi }{4}$ between 0 and $2\pi =\frac{8\pi }{4}$. Subtract $2\cdot 2\pi =\frac{16\pi }{4}$ to get $\frac{17\pi }{4}-\frac{16\pi }{4}=\frac{\pi }{4}$. $\frac{\pi }{4}$ lies in Quadrant 1, where sine is positive. The exact value of $\sin \left(\frac{\pi }{4}\right)$ is $\frac{\sqrt{2}}{2}$. Options B, C, and D can be eliminated for incorrect magnitude or sign. Correct answer: A.

Question 2 (Free Response)

Consider angle $\theta$ with terminal side passing through the point $(-2,3)$ on the coordinate plane. (a) Find the length $r$ from the origin to the point $(-2,3)$. (b) What are the exact values of $\cos \theta$ and $\sin \theta$? (c) If $\varphi$ is coterminal with $\theta$ and $0\le \varphi <2\pi$, what quadrant is $\varphi$ in, and what is the reference angle for $\varphi$?

Worked Solution: (a) Use the distance formula from the origin: $$r=\sqrt{(-2{)}^2+3^2}=\sqrt{4+9}=\sqrt{13}$$ So the length $r=\sqrt{13}$.

(b) For any point $(x,y)$ at distance $r$ from the origin, the intersection with the unit circle is $\left(\frac{x}{r},\frac{y}{r}\right)$, so: $$\cos \theta =\frac{x}{r}=\frac{-2}{\sqrt{13}}=-\frac{2\sqrt{13}}{13}\text{and}\sin \theta =\frac{y}{r}=\frac{3}{\sqrt{13}}=\frac{3\sqrt{13}}{13}$$

(c) The x-coordinate of $\theta$ is negative and the y-coordinate is positive, so $\varphi$ is in Quadrant II. For a Quadrant II angle, the reference angle $\alpha =\pi -\varphi$, so $\alpha =\pi -\arccos \left(\frac{2\sqrt{13}}{13}\right)$.

Question 3 (Application / Real-World Style)

A Ferris wheel with radius 10 meters has its center 15 meters above the ground. A rider starts at the 3 o'clock position (same height as the center, right of the center). The wheel rotates counterclockwise by 210 degrees. What is the rider's height above the ground after this rotation, to the nearest tenth of a meter?

Worked Solution: First convert 210 degrees to radians: $210^{\circ }=\frac{7\pi }{6}$ radians. The vertical position of the rider relative to the center of the wheel is $r\sin \theta$, where $r=10$ meters. From unit circle values, $\sin \left(\frac{7\pi }{6}\right)=-\frac{1}{2}$. The relative vertical position is $10\cdot \left(-\frac{1}{2}\right)=-5$ meters, meaning the rider is 5 meters below the center. Add the center's height above ground: $15+(-5)=10.0$ meters. In context, after rotating 210 degrees counterclockwise from the starting position, the rider is 10.0 meters above the ground.

7. Quick Reference Cheatsheet

8. What's Next

This topic is the foundational building block for all remaining trigonometric and polar topics in Unit 3. Immediately next, you will use unit circle sine and cosine values to graph sine and cosine functions, identify their amplitude, period, and phase shift, and model periodic real-world phenomena like seasonal temperature variation or tidal motion. Without mastering exact unit circle values and sign rules, you will not be able to correctly evaluate trigonometric functions at key points, find intercepts and extrema of trig graphs, or convert between rectangular and polar coordinates later in the unit. This topic also underpins future work with trigonometric identities and inverse trigonometric functions, which are tested heavily on the AP Precalculus exam.

• Graphs of sine and cosine functions
• Polar coordinates and polar graphs
• Pythagorean and reciprocal trigonometric identities
• Inverse sine and inverse cosine functions