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Quizzes > Mathematics

Unit Circle Quiz: Practice Radians, Degrees, and Key Coordinates

Quick unit circle practice test. 20 questions, instant results.

Editorial: Review CompletedCreated By: Abhi JithUpdated Aug 27, 2025
Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art promoting a trivia quiz about the unit circle for high school trigonometry students.

This unit circle quiz helps you practice radians, degrees, signs, and exact sine and cosine at common angles. Work through 20 quick questions to check what you know and build speed. Want more? Try the unit circle values quiz, drill coordinates with fill in the unit circle, or broaden your skills with an interactive trigonometry quiz.

What is the radius of the unit circle?
-1
1
2
0
The definition of a unit circle is that it has a radius of 1. This fact forms the basis for all trigonometric evaluations on the unit circle.
Convert 90 degrees to radians.
Ï€/3
2Ï€/3
Ï€
Ï€/2
To convert degrees to radians, multiply the degree measure by Ï€/180. Thus, 90° equals 90 Ã- (Ï€/180) = Ï€/2 radians.
What are the coordinates of the point at 0° on the unit circle?
(0, -1)
(0, 1)
(1, 0)
(-1, 0)
At 0°, the point lies on the positive x-axis, which corresponds to the coordinates (1, 0). This is a fundamental reference point on the unit circle.
What is the sine of 90° on the unit circle?
0
 
-1
1
Sine corresponds to the y-coordinate on the unit circle. At 90°, the point is (0, 1), so the sine value is 1.
What is the cosine of 0° on the unit circle?
-1
 
0
1
Cosine represents the x-coordinate in the unit circle. At 0°, the coordinate is (1, 0), so the cosine value is 1.
What is the degree measure of π/3 radians?
60 degrees
45 degrees
90 degrees
30 degrees
Converting radians to degrees involves multiplying the radian measure by 180/π. Therefore, π/3 radians equals 60 degrees.
Which of the following points is located in the second quadrant of the unit circle?
(√2/2, √2/2)
(√2/2, -√2/2)
(-√2/2, -√2/2)
(-√2/2, √2/2)
The second quadrant is characterized by a negative x-coordinate and a positive y-coordinate. The point (-√2/2, √2/2) meets these criteria.
What is the tangent of 45° on the unit circle?
1
-1
 
0
At 45°, sine and cosine have the same value, making the tangent - defined as sine divided by cosine - equal to 1. This is a classic result in trigonometry.
Using the coordinates (cos θ, sin θ), what is the value of sin²θ + cos²θ?
sin θ
cos θ
0
1
The Pythagorean identity in trigonometry states that sin²θ + cos²θ = 1 for any angle θ. This identity is true for all points on the unit circle.
How many degrees are there in a full rotation around the unit circle?
2Ï€ degrees
180 degrees
90 degrees
360 degrees
A complete revolution around a circle measures 360 degrees. This is a universally accepted fact in geometry.
Which angle in radians corresponds to the point where cosine equals 0 and sine equals 1?
0
Ï€/2
Ï€
3Ï€/2
The point (0, 1) on the unit circle occurs at π/2 radians, where the cosine is 0 and the sine is 1. This is an essential angle in trigonometric studies.
What is the relationship between an angle's radian measure and its corresponding arc length on the unit circle?
Arc length equals twice the angle
Arc length equals half the angle
Arc length equals the angle in radians
Arc length equals the square of the angle
On a unit circle, the arc length for a given angle is numerically equal to the angle when measured in radians. This direct relationship is a fundamental property of the circle.
Which of the following is the correct radian conversion for 270°?
Ï€/2
Ï€
3Ï€/2
2Ï€
To convert 270° to radians, multiply 270 by π/180 resulting in 3π/2. This is a standard conversion in trigonometry.
In which quadrant are cosine values negative while sine values remain positive?
Second Quadrant
First Quadrant
Fourth Quadrant
Third Quadrant
In the second quadrant, the x-values (cosine) are negative and the y-values (sine) are positive. This helps in identifying the signs of trigonometric functions based on the angle's location.
Which trigonometric function is undefined at 90° on the unit circle?
Cosine
Tangent
Sine
Secant
Tangent is calculated as sine divided by cosine. At 90°, cosine is 0, which makes the tangent undefined. This is an important concept in trigonometry.
If cos θ = 1/2 on the unit circle, what are the possible measures of θ in the interval [0, 2π)?
Ï€/4 and 3Ï€/4
Ï€/3 and 5Ï€/3
Ï€/6 and 5Ï€/6
2Ï€/3 and 4Ï€/3
Cosine equals 1/2 at two points in the interval [0, 2π): π/3 and 5π/3. These are the only solutions in one full rotation where the cosine value is 1/2.
What is the exact value of sin(7Ï€/6) on the unit circle?
-1/2
1/2
√3/2
-√3/2
The angle 7π/6 is in the third quadrant where sine is negative, and its reference angle is π/6 whose sine is 1/2. Therefore, sin(7π/6) equals -1/2.
Determine the coordinates of the point on the unit circle corresponding to an angle of 225°.
(-√2/2, -√2/2)
(√2/2, √2/2)
(-√2/2, √2/2)
(√2/2, -√2/2)
225° converts to 5π/4 radians, which places the point in the third quadrant where both coordinates are negative. Therefore, the coordinates are (-√2/2, -√2/2).
If the tangent of an angle is √3, which of the following could be that angle in radians?
Ï€/4
2Ï€/3
Ï€/3
Ï€/6
Tangent is the ratio of sine to cosine, and at π/3 radians the tangent equals √3. This is a well-known property of the 60° angle in trigonometry.
Which of the following represents the correct Pythagorean identity on the unit circle?
sin θ + cos θ = 1
sin²θ + cos²θ = 1
sin²θ - cos²θ = 1
1 - sin²θ = cos θ
The fundamental Pythagorean identity in trigonometry is sin²θ + cos²θ = 1, which holds true for every angle on the unit circle. This identity is crucial for many trigonometric proofs and applications.
0
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Study Outcomes

  1. Understand the definitions and properties of the unit circle.
  2. Apply the relationships between angles and their corresponding coordinates.
  3. Calculate sine, cosine, and tangent values for standard angles.
  4. Analyze the behavior of trigonometric functions across different quadrants.
  5. Interpret the significance of radian measure on the unit circle.

Unit Circle Quiz: Practice Test Cheat Sheet

  1. Unit Circle Basics - Imagine a circle of radius 1 centered at the origin; this simple shape is the foundation of all trigonometry. Every point on this circle corresponds to an angle and gives you sine and cosine values in a snap.
  2. SOH‑CAH‑TOA Mnemonic - Lock in your sine, cosine, and tangent ratios with the classic "SOH‑CAH‑TOA" chant: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. This catchy phrase will save you brainpower during crunch time.
  3. ASTC Quadrant Signs - Use "All Students Take Calculus" to remember sign rules in each quadrant: all positive in I, sine positive in II, tangent positive in III, and cosine positive in IV. This quick trick stops you from second‑guessing signs mid‑problem.
  4. Key Angle Values - Memorize sin and cos for 0°, 30°, 45°, 60°, and 90° using the pattern sin(θ)=√n/2 and cos(θ)=√(4−n)/2 (n=0,1,2,3,4). This formulaic approach helps you recall exact values without a calculator.
  5. Left‑Hand Trick - Bend each finger on your left hand to represent 0°, 30°, 45°, 60°, and 90°; the number of fingers below the bend gives sin, and above gives cos (take √finger count/2). This hands‑on hack turns abstract numbers into a fun handshake.
  6. Beyond Right Triangles - The unit circle extends trig functions to all real angles, not just acute ones in right triangles. Embrace this broader view to solve problems that go past 0° - 90°.
  7. Coordinates as Cos & Sin - On the unit circle, every point's x‑coordinate equals cos(θ) and y‑coordinate equals sin(θ). This geometric link is your shortcut to visualizing and calculating values.
  8. Degrees ↔ Radians - Convert like a pro: 180° equals π radians, so multiply or divide by π/180 to switch between units. Mastering this interchange keeps your answers in the right format.
  9. Reciprocal Functions - Expand your toolkit with csc = 1/sin, sec = 1/cos, and cot = 1/tan. These reciprocals pop up in identities and integrals, so get comfy with them now.
  10. Trig Identity Mnemonics - Use playful phrases like "Some Old Horses Chew Apples Happily Throughout Old Age" to nail SOH‑CAH‑TOA and other identities. Mnemonics turn dry formulas into memorable jingles.
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