Understanding the Unit Circle
The unit circle is a special kind of circle that has a radius of 1 and is centered at the origin (0,0) on a coordinate plane. It might not seem like much at first glance, but this simple circle is the key to understanding a whole host of important concepts in pre-calculus, from trigonometry to polar coordinates.
To begin, we need to understand the concept of an angle in standard position. An angle in standard position is an angle measured starting from the positive x-axis and going counterclockwise.
Above is an image of the unit circle. It may look very complicated at first glance, but you'll get very familiar with the unit circle in this unit, and you will realize that it's not that complicated after all. This circle has a radius of 1 unit on the coordinate plane -- that's why the points at which it intersects the x- and y- axes are variations of the point (1,0). We look at the unit circle starting not at the top, but at the right, on the positive x-axis. Here, we can see that the angle that is written on the circle is 0 degrees (or 0 radians).
Finding Angles on the Unit Circle
To make a full rotation of the unit circle, we start at 0 degrees and look counterclockwise until we get back to the positive x-axis. As you go around the circle, there are values written in radians and their corresponding values in degrees on the inside of the circle. These values are the measures of the angle the ray they are on makes with the positive x-axis, measured in standard position.
For example, in radians represents that the angle made by the ray it lies on and the positive x-axis is radians, or 30 degrees. 300 degrees, or radians, represents that the angle the ray it lies on makes with the positive x-axis is 300 degrees when measured counterclockwise from the positive x-axis. We can see that this is a reflex angle (an angle that measures greater than 180 degrees but less than 360 degrees) and not the most efficient way to measure this angle. An angle of radians counterclockwise is the same as an angle of radians. This angle is now negative because we are measuring it clockwise from the positive x-axis, so we are going down to , rather than all the way around the circle. However, both and still identify the same angle on the unit circle.
Another important feature of the unit circle is that one full revolution around it is equal to radians, or 360 degrees. You can revolve around the unit circle as many times as you want, increasing the value of your angle by radians each time. For example, if you were to make 3 and a half revolutions of the unit circle, you would multiply by 3, yielding , then add a to that for the half-circle. You will have covered radians in 3 and a half revolutions.
Trigonometry with the Unit Circle
When we take an angle in standard position and draw a line (called the "terminal ray") that goes out from the origin at that angle, the point where this line intersects the unit circle is called Point P. The coordinates of Point P are given by , where is the measure of the angle in radians or degrees. These coordinates are based on the trigonometric functions sine and cosine, which are used to describe the relationships between the angles and sides of a right triangle. The sine of an angle is the y-coordinate of the point on the unit circle, and the cosine is the x-coordinate. The tangent of an angle is the ratio of the y-coordinate to the x-coordinate, or .
To find the sine, cosine, or tangent values at a specific angle, we can simply use the coordinates of Point P. For example, if we want to find the sine of 30 degrees, we can draw the angle in standard position on the unit circle, and find the point of intersection, P. The y-coordinate of P is the sine value at 30 degrees, which is 0.5. Similarly, the x-coordinate of P is the cosine value at 30 degrees, which is . And the tangent value at 30 degrees is .
It's important to note that these values remain constant regardless of the size of the right triangle. These values are called the "unit circle's ratios" and they can be used to find the values of sine, cosine and tangent for any angle.
You can use the acronym SOHCAHTOA to remember how to find the sine, cosine, and tangent of an angle when given a right triangle. SOH = Sine is the value of the side Opposite the angle over the value of the Hypotenuse. CAH = Cosine is the value of the side Adjacent to the angle over the Hypotenuse. TOA = Tangent is the side Opposite the angle over the Hypotenuse.
As you can see in the image above, the unit circle is created using the side lengths of the right triangles created by various angles. The hypotenuse of a triangle on the unit circle will always be 1 because that is the radius of the circle. The horizontal side of the triangle is used to find the cosine of the angle θ, so it is the x-coordinate of point P. The vertical side of the triangle is used to find the sine of the angle θ, so it is the y-coordinate of point P.
Finding the Angle Given the Trig Value
Using the unit circle, we can also find the angle given the sine or cosine value. If we are given that the cosine of the angle we are looking for is 1/2, we can find where 1/2 is an x-coordinate on the unit circle, since we know that cosine values are the x-coordinates. Starting at the 0-degree mark, or the positive x-axis, we can work our way around the unit circle counterclockwise until we find an x-coordinate of 1/2. Very quickly, we would see that 1/2 is an x-coordinate when the angle is radians, or 60 degrees.
If we are given that the sine of the angle is and are asked to find the angle, we would follow a similar process as before. Starting at the positive x-axis and moving counterclockwise, we now look at the y-values on the unit circle to find . We find what we are looking for in the 3rd quadrant (bottom left) of the coordinate plane, at an angle of radians.
Determining the Sign of Trig Values
An important concept to learn is identifying when these trigonometric functions are positive or negative.
The acronym "All Students Take Calculus" is useful for remembering when each trigonometric function is positive.
All: All the trig values (sine, cosine, and tangent) are positive in the first quadrant.
Students: The Sine values are positive in the second quadrant, and all the other values are negative.
Take: The Tangent values are positive in the third quadrant, and all the other values are negative.
Calculus: The Cosine values are positive in the fourth quadrant, and all the other values are negative.
Frequently Asked Questions
How do I find the coordinates of a point on a circle using sine and cosine?
Use the formula from the CED: for a circle centered at the origin with radius r and an angle θ in standard position, the intersection point P has coordinates (r cos θ, r sin θ). On the unit circle (r = 1) that’s just (cos θ, sin θ). How to get numbers: - Find the reference angle (acute angle between the terminal ray and the x-axis). - Use known exact values from 30°–60°–90° and 45°–45°–90° triangles (e.g. cos 30° = √3/2, sin 45° = √2/2). - Apply the correct signs based on the quadrant (I: +,+; II: −,+; III: −,−; IV: +,−). Quick example: θ = 150° = 5π/6. Reference angle = 30°, so cos = −√3/2, sin = +1/2. For radius r, multiply: P = (r(−√3/2), r(1/2)). For more examples and AP-aligned practice, check the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and extra problems (https://library.fiveable.me/practice/ap-pre-calculus). Remember: on the AP exam use radian mode when a calculator is required.
What's the formula for finding points on a circle when I know the angle and radius?
If you have a circle centered at the origin with radius r and an angle θ in standard position, the terminal ray hits the circle at the point P = (r cos θ, r sin θ). For the unit circle (r = 1) that’s (cos θ, sin θ). Remember to use the reference angle and the quadrant to get the correct signs for cos and sin (±). For exact values at common angles use 30°/60°/45° triangles (π/6, π/3, π/4) and pay attention if the angle is quadrantal (on an axis). On the AP exam your calculator should be in radian mode when required; you’ll often convert between radians and degrees for exact values. For more practice and AP-aligned review see the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH), the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3), and plenty of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
When do I use r cos θ and r sin θ to find coordinates?
Use r cos θ and r sin θ whenever you have an angle θ in standard position and a circle (or polar point) centered at the origin with radius r. The terminal ray of θ hits the circle at P with coordinates (x,y) = (r cos θ, r sin θ). For the unit circle r = 1 this simplifies to (cos θ, sin θ). Keep these practical points in mind: - Signs depend on the quadrant of θ (use CAST or check cos/sin sign). - For exact values at multiples of π/6 and π/4, build 30-60-90 and isosceles right triangles and apply the quadrant sign (CED 3.3.A.1–A.2). - In polar-to-rectangular conversion use x = r cos θ, y = r sin θ. - On the AP exam keep your calculator in radian mode for calculator sections (Part A of Section II uses radian mode). For a quick AP-aligned review, see the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH). For extra practice, check the unit overview (https://library.fiveable.me/ap-pre-calculus/unit-3) and the practice bank (https://library.fiveable.me/practice/ap-pre-calculus).
I'm confused about why the x-coordinate is r cos θ and the y-coordinate is r sin θ - can someone explain?
Think of the angle θ in standard position with its vertex at the origin and the terminal ray hitting the circle of radius r at point P. Drop a perpendicular from P to the x-axis—that makes a right triangle with hypotenuse r and acute angle θ (the reference angle). By basic right-triangle trig: - adjacent side = r · cos θ, which is the horizontal distance from the origin to P → x = r cos θ. - opposite side = r · sin θ, which is the vertical distance from the origin to P → y = r sin θ. So P = (r cos θ, r sin θ). The signs come from which quadrant the terminal ray is in (cos and sin can be negative depending on x or y direction). This is exactly the CED statement (3.3.A.1): coordinates on a circle centered at the origin are (r cos θ, r sin θ). For more review and examples (including multiples of π/4 and π/6 with triangle geometry and sign rules), check the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) or the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3). For extra practice, use the AP Precalculus problem set (https://library.fiveable.me/practice/ap-pre-calculus).
How do I find exact values for sine and cosine of π/4 and π/6 without a calculator?
Think “unit circle + nice triangles.” For exact values without a calculator, draw the unit circle (r = 1) and use the relevant reference triangle geometry. - π/4 (45°): use an isosceles right triangle with legs 1,1 → hypotenuse √2. On the unit circle scale both legs by 1/√2 to get coordinates (cos θ, sin θ) = (√2/2, √2/2). So cos(π/4) = √2/2 and sin(π/4) = √2/2. - π/6 (30°) and π/3 (60°): use a 30–60–90 triangle with sides 1 (short), √3 (long), 2 (hypotenuse). Scale to unit hypotenuse: divide by 2 → short = 1/2, long = √3/2. For 30° (π/6) the coordinates are (cos, sin) = (√3/2, 1/2). So cos(π/6)=√3/2, sin(π/6)=1/2. For π/3 swap those values. Remember to adjust signs by quadrant (use the terminal ray/reference angle idea from the CED). On the AP exam you must show this geometric work for no-calculator FRQs. For more examples and practice, check the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and extra practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
What's the difference between finding coordinates on a unit circle vs a circle with radius r?
Short answer: on the unit circle (r = 1) the point where the terminal ray at angle θ hits the circle is (cos θ, sin θ). For any circle centered at the origin with radius r, that same ray hits at (r cos θ, r sin θ)—you just scale both coordinates by r (CED 3.3.A.1). Why that matters: cosine and sine always give the x- and y- coordinates of the direction of the ray (from the unit circle). If r ≠ 1, multiply those values by r to get the actual coordinates. Signs still depend on the quadrant; exact values for common angles come from 45° and 30–60–90 triangles (CED 3.3.A.2). On the AP exam remember angle measures are in radians unless told otherwise and use exact values when possible (no calculator for some FRQ parts). For more review and worked examples, see the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
How do I use special right triangles to find exact trig values?
Use the two special right triangles and the unit circle + reference angles. - 45° (π/4) is from an isosceles right triangle: legs 1,1, hypotenuse √2. On the unit circle divide by hypotenuse → cos(π/4)=sin(π/4)=1/√2 = √2/2. - 30° (π/6) and 60° (π/3) come from a 30–60–90 triangle with sides 1 (short), √3 (long), 2 (hyp). On the unit circle: cos(π/6)=√3/2, sin(π/6)=1/2; cos(π/3)=1/2, sin(π/3)=√3/2. Process: find the reference angle (acute angle between terminal ray and x-axis), use the appropriate triangle value, then apply the sign based on the quadrant (cos from x-coordinate, sin from y-coordinate). For example, for 5π/4 the reference is π/4 but both sine and cosine are negative, so sin(5π/4)=−√2/2, cos(5π/4)=−√2/2. AP tip: the exam expects exact values (no decimals) for multiples of π/4 and π/6—use (cos θ, sin θ) on the unit circle and quadrant signs. For extra practice and quick reference, see the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and hundreds of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
When the angle is in different quadrants, how do I know if sine and cosine are positive or negative?
Think of the unit circle: a point at angle θ has coordinates (cos θ, sin θ)—so the sign of cosine is the x-coordinate sign and the sign of sine is the y-coordinate sign (CED 3.3.A.1). That gives the quadrant rule: - Quadrant I (0 to π/2): x>0, y>0 → cos+, sin+ - Quadrant II (π/2 to π): x<0, y>0 → cos−, sin+ - Quadrant III (π to 3π/2): x<0, y<0 → cos−, sin− - Quadrant IV (3π/2 to 2π): x>0, y<0 → cos+, sin− If the terminal ray lies on an axis (quadrantal angle), one of the values is 0. For exact values (multiples of π/6 or π/4), use reference angles and the appropriate signs from the quadrant (CED 3.3.A.2). Practice unit-circle work for fluency—see the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH), the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3), and lots of practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
Can someone explain step by step how to find the coordinates where the terminal ray intersects the circle?
Start with the CED fact: if a circle is centered at the origin with radius r and an angle θ is in standard position, the terminal ray meets the circle at P = (r cos θ, r sin θ). Step-by-step: 1. Put θ in standard position (vertex at origin, initial side on +x-axis). 2. If r ≠ 1, work with radius r; for the unit circle use r = 1 so P = (cos θ, sin θ). 3. Find the reference angle (acute angle to the x-axis) and the exact cos/sin from 30°, 45°, 60° triangles: - 30° (π/6): (√3/2, 1/2) - 45° (π/4): (√2/2, √2/2) - 60° (π/3): (1/2, √3/2) 4. Determine the quadrant of θ to set signs: (I:+,+) (II:-,+) (III:-,-) (IV:+,-). 5. Multiply those sine/cosine values by r if r ≠ 1. Example: θ = 5π/6 (quadrant II), reference = π/6 → cos = -√3/2, sin = +1/2 → P = (r(-√3/2), r(1/2)). For more practice and AP-aligned review, see the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH). Remember the AP exam expects radian mode when using calculators.
I don't understand how isosceles right triangles help me find sine and cosine values - what's the connection?
Think of the unit circle as a bunch of right triangles with hypotenuse 1. An isosceles right triangle (45°–45°–90°) has legs of length 1/√2 = √2/2. So when the terminal ray makes a 45° (π/4) angle, the point on the unit circle is (cos θ, sin θ) = (√2/2, √2/2). That gives the exact sine and cosine for ±45° angles. For other quadrants, keep the same reference triangle but change signs based on x (cos) and y (sin) being positive or negative. The same idea applies to 30°–60° (30°–60°–90°) triangles: leg ratios 1 : √3 : 2 give cos 30° = √3/2, sin 30° = 1/2, etc. On the AP exam you’ll often need these exact, no-calculator values and to write coordinates as (r cos θ, r sin θ) per the CED. For extra practice and a quick reference, check the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and the unit practice set (https://library.fiveable.me/practice/ap-pre-calculus).
How do I memorize the exact values for sine and cosine of common angles like π/6, π/4, and π/3?
Think of the unit circle and two special triangles—the 45° isosceles right (π/4) and the 30–60–90 (π/6 and π/3). Put the triangle on the unit circle so the point is (cos θ, sin θ). For π/4: the 45–45–90 gives legs 1, hyp √2, so cosπ/4 = sinπ/4 = √2/2. For π/6 (30°) and π/3 (60°): the 30–60–90 has sides 1 (short), √3 (long), 2 (hyp). On the unit circle divide by 2 → for 30°: cosπ/6 = √3/2, sinπ/6 = 1/2. For 60° swap them: cosπ/3 = 1/2, sinπ/3 = √3/2. Use reference angles and quadrant signs: coordinates = (cos θ, sin θ) and change signs depending on quadrant. On the AP exam you’ll give exact values (no calculator) and work in radian mode. For a quick refresher, check the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and try practice problems (https://library.fiveable.me/practice/ap-pre-calculus).
What are the coordinates of a point on a circle with radius 5 at angle π/3?
Use the formula from the CED: for a circle radius r centered at the origin, a point at angle θ has coordinates (r cos θ, r sin θ). Here r = 5 and θ = π/3. Cos(π/3) = 1/2 and sin(π/3) = √3/2, so the coordinates are (5·cos(π/3), 5·sin(π/3)) = (5/2, (5√3)/2). This point lies in Quadrant I (both coordinates positive). If you want more on using reference angles and exact sine/cosine values for π/6, π/4, π/3, check the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH). For extra practice problems, see the AP Precalculus practice bank (https://library.fiveable.me/practice/ap-pre-calculus).
Why do the signs of sine and cosine change in different quadrants and how do I remember which is positive where?
Sine and cosine signs come from the point coordinates on a circle: for angle θ the terminal point is (r cos θ, r sin θ) (CED 3.3.A.1). On the unit circle r = 1, so cos θ is the x-coordinate and sin θ is the y-coordinate. Signs change because x and y change sign in different quadrants: - Quadrant I: x > 0, y > 0 → cos +, sin + - Quadrant II: x < 0, y > 0 → cos −, sin + - Quadrant III: x < 0, y < 0 → cos −, sin − - Quadrant IV: x > 0, y < 0 → cos +, sin − Use reference angles and 30°/45°/60° triangle geometry (CED 3.3.A.2) to get exact values, then apply the quadrant signs. A handy mnemonic: “All Students Take Calculus” (ASTC)—All (+) in QI, Sine (+) in QII, Tangent (+) in QIII, Cosine (+) in QIV. For extra practice and AP-aligned examples, see the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and Unit 3 resources (https://library.fiveable.me/ap-pre-calculus/unit-3). For more problems, try the practice set (https://library.fiveable.me/practice/ap-pre-calculus).
How do I solve problems where I need to find both the angle and the coordinates on a circle?
Start by finding the angle θ, then use (r cos θ, r sin θ) for the coordinates. Steps: 1. Put the angle in standard position (initial side on +x). If you’re given a trig value, find the reference angle α using inverse trig, then locate the correct quadrant to get θ (watch signs of sine/cosine). 2. If θ is a common multiple of π/6 or π/4, use 30–60–90 and isosceles right triangles to get exact cos and sin values (remember to apply +/− by quadrant). 3. For a circle of radius r, compute x = r·cos θ and y = r·sin θ. On the unit circle (r = 1) those are just (cos θ, sin θ). 4. On the AP exam do exact values when possible; no calculator is allowed for many trig free-response parts, and calculators should be in radian mode for Part A (CED). Practice these steps with problems in the Topic 3.3 study guide (https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH) and more problems at (https://library.fiveable.me/practice/ap-pre-calculus).
I keep getting confused about when to use 30-60-90 triangles vs 45-45-90 triangles for exact trig values - when do I use which one?
Use the triangle that matches the reference angle. If the reference angle is 45° (π/4) use a 45–45–90 isosceles right triangle; if it’s 30° (π/6) or 60° (π/3) use a 30–60–90 equilateral-based triangle. Put that triangle in the unit circle (r = 1) so coordinates are (cos θ, sin θ), then pick signs based on the quadrant of the terminal ray (CED 3.3.A.1–A.2). Quick exact values to memorize: - 45° (π/4): sin = ±√2/2, cos = ±√2/2 (same magnitude). - 30° (π/6): sin = 1/2, cos = √3/2. - 60° (π/3): sin = √3/2, cos = 1/2. Tip: convert any angle to its reference angle (acute) and then apply the triangle values, flipping signs by quadrant. On the AP exam, these exact values show up often on no-calculator parts—practice doing them with the unit circle (see the Topic 3.3 study guide: https://library.fiveable.me/ap-pre-calculus/unit-3/sine-cosine-function-values/study-guide/lz6lqowpANg0eU40lNHH). For more review, check the Unit 3 overview (https://library.fiveable.me/ap-pre-calculus/unit-3) and practice problems (https://library.fiveable.me/practice/ap-pre-calculus).