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Quizzes > Mathematics

Unit Circle Practice Quiz: Degrees, Radians & Coordinates

Quick unit circle radians quiz to check your skills. Instant results and explanations.

Editorial: Review CompletedCreated By: Evron GultomUpdated Aug 28, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration for a unit circle quiz on a golden yellow background

Use this quiz to sharpen your unit circle skills, from degrees and radians to sine, cosine, and key coordinates. For more targeted practice, try our unit circle practice test or a hands-on unit circle fill in quiz, or broaden your review with an interactive trigonometry quiz.

What are the coordinates of the point on the unit circle at angle 0 radians?
(-1, 0)
(0, -1)
(0, 1)
(1, 0)
On the unit circle, the angle 0 radians corresponds to the point (1, 0). The cosine of 0 is 1 and the sine is 0, giving the coordinates (cos0, sin0) = (1, 0). This is the standard starting point for measuring angles in the unit circle.
What is the value of sin(?/2)?
-1
?2/2
1
0
At ?/2 radians (90°), the unit circle point is (0,1) so sine - which corresponds to the y-coordinate - is 1. This is a fundamental value of the sine function at a right angle. Remember that sine represents vertical height on the unit circle.
What is the value of cos(?)?
0
?3/2
-1
1
At ? radians (180°), the unit circle point is (-1,0), so cosine - which corresponds to the x-coordinate - is -1. Cosine measures the horizontal projection from the origin. This is a key quadrantal value for cosine.
What is tan(?/4)?
?3
?2
1
0
At ?/4 (45°), sine and cosine are both ?2/2, so tan - sine over cosine - is (?2/2)/(?2/2) = 1. Tangent measures the slope of the radius line. This is a classic symmetry on the unit circle. https://en.wikipedia.org/wiki/Tangent_function
What is sin(?/6)?
1
?2/2
1/2
?3/2
At ?/6 (30°), the unit circle coordinate is (?3/2,1/2), so sine - the y-coordinate - is 1/2. This value is one of the fundamental angles in trigonometry. Memorizing these half-angle values helps with many calculations.
What is cos(?/3)?
?2/2
-1/2
?3/2
1/2
At ?/3 (60°), the unit circle coordinate is (1/2, ?3/2), so cosine - the x-coordinate - is 1/2. This is one of the standard values used in many trigonometric problems. Knowing this helps with geometry and wave analysis.
What is tan(?/6)?
1
?2/2
?3
1/?3
At ?/6 (30°), sine is 1/2 and cosine is ?3/2, so tan = (1/2)/(?3/2) = 1/?3. Tangent measures the ratio of vertical to horizontal components on the unit circle. Recognizing this value simplifies angle calculations. https://en.wikipedia.org/wiki/Tangent_function
Which angle in radians is equivalent to 90°?
?/3
?/2
3?/2
2?
90 degrees corresponds to a quarter turn around the unit circle, which is ?/2 radians. Converting degrees to radians uses the ratio ? radians = 180 degrees. Thus 90° × (?/180) = ?/2.
What is cos(2?/3)?
-1/2
1/2
-?3/2
?3/2
At 2?/3 (120°), the unit circle coordinate is (-1/2, ?3/2), so cosine = -1/2. Cosine is negative in the second quadrant. Knowing quadrant signs helps determine correct values. https://en.wikipedia.org/wiki/Unit_circle#Signs_of_trigonometric_functions
What is sin(3?/4)?
?2/2
-?2/2
?3/2
1/2
At 3?/4 (135°), the unit circle coordinate is (-?2/2, ?2/2), so sine = ?2/2. Sine is positive in the second quadrant. Recognizing these 45°-based angles is crucial for trig problems.
What is tan(5?/4)?
1
-?3
?3
-1
At 5?/4 (225°), the unit circle coordinate is (-?2/2, -?2/2), so tan = (-?2/2)/(-?2/2) = 1. Both sine and cosine are negative in the third quadrant, making their ratio positive. https://en.wikipedia.org/wiki/Tangent_function
What are the coordinates at angle 3?/2?
(1, 0)
(-1, 0)
(0, -1)
(0, 1)
At 3?/2 (270°), the unit circle coordinate is (0, -1). Cosine gives the x-coordinate (0) and sine gives the y-coordinate (-1). This is a standard quadrantal point.
Which angle has coordinates (-?3/2, -1/2)?
2?/3
4?/3
5?/6
7?/6
The point (-?3/2, -1/2) lies in the third quadrant where both cosine and sine are negative. 4?/3 (240°) yields coordinates (??3/2, ?1/2). Recognizing quadrant signs and reference angles leads to this answer.
What is the reference angle for 5?/3?
?/2
?/6
2?/3
?/3
5?/3 (300°) is in the fourth quadrant. The reference angle is found by subtracting from 2?: 2? ? 5?/3 = ?/3. Reference angles are always acute and help determine sine and cosine magnitudes.
What is sin(7?/6)?
?3/2
-?3/2
1/2
-1/2
At 7?/6 (210°), the unit circle coordinate is (-?3/2, -1/2), so sine = -1/2. This angle is in the third quadrant where sine is negative. Recognizing standard angles helps evaluation.
What is cos(4?/3)?
1/2
-1/2
?3/2
-?3/2
At 4?/3 (240°), the unit circle coordinate is (-1/2, -?3/2), so cosine = -1/2. In the third quadrant, cosine values are negative. Knowing these key points speeds up trigonometric calculations.
What is tan(11?/6)?
1/?3
?3
-1/?3
-?3
At 11?/6 (330°), the point is (?3/2, -1/2), so tan = (-1/2)/(?3/2) = -1/?3. The angle is in the fourth quadrant where tangent is negative. Simplifying the fraction gives the exact value. https://en.wikipedia.org/wiki/Tangent_function
Which angle between 0 and 2? has sin = -?3/2 and cos positive?
5?/3
11?/6
4?/3
7?/6
Sine = -?3/2 means a reference angle of ?/3 with negative y-coordinate. Cosine must be positive, so the angle is in the fourth quadrant: 2? ? ?/3 = 5?/3. Recognizing quadrant signs is essential for selecting the correct solution.
What is the smallest positive angle less than 2? at which the tangent function is undefined?
?
3?/2
2?
?/2
Tangent is sine over cosine and is undefined when cosine = 0. The first positive quadrantal angle where cos = 0 is ?/2. This is the smallest positive value where tan is undefined before 2?. https://en.wikipedia.org/wiki/Tangent_function
Which angle in the second quadrant has cos = -?2/2?
5?/4
?/4
3?/4
7?/4
Cosine = -?2/2 indicates a reference angle of ?/4 in a quadrant where cosine is negative. The second quadrant angle with reference ?/4 is ? ? ?/4 = 3?/4. Recognizing reference angles and sign conventions is key.
Evaluate cos(15?/4).
-?2/2
1/2
-1/2
?2/2
15?/4 reduces by subtracting 2? (8?/4): 15?/4 ? 8?/4 = 7?/4. Cos(7?/4) = ?2/2. Simplifying angles modulo 2? yields equivalent values on the unit circle.
What is the principal value of arcsin(-?2/2)?
-?/4
3?/4
-3?/4
?/4
Arcsin returns values in [??/2, ?/2]. Since sine is -?2/2 at -?/4, the principal value is -?/4. This avoids angles outside the function's defined range. https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
Given the point (-1/2, ?3/2) on the unit circle, what is the corresponding angle in [0,2?)?
4?/3
2?/3
?/3
5?/3
The coordinates (-1/2, ?3/2) lie in the second quadrant, where cosine is negative and sine is positive. The reference angle is ?/3, so the angle is ? ? ?/3 = 2?/3. Identifying the correct quadrant is essential.
What is the exact value of cos(5?/12)?
(?6 + ?2)/4
(?6 + ?3)/4
(?3 ? ?2)/4
(?6 ? ?2)/4
5?/12 is 75°, which can be expressed as ?/3 + ?/12 or ?/4 + ?/6, allowing use of sum formulas. Cos(5?/12) = cos(?/3 + ?/12) or via half-angle identity yields (?6 + ?2)/4. This exact radical form comes from combining known angles.
What is the exact value of tan(7?/12)?
2+?3
-(2+?3)
-(2??3)
2??3
7?/12 is 105°, which equals ?/4 + ?/6. Using the tangent sum formula tan(A+B) = (tanA+tanB)/(1?tanA·tanB) with tan(?/4)=1 and tan(?/6)=1/?3 yields -(2+?3). This exact form arises from simplifying that expression.
0
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Study Outcomes

  1. Identify key unit circle angles -

    Quickly recognize and convert between common angles in degrees and radians on the unit circle quiz to build a strong foundation for unit circle practice.

  2. Recall exact radian measures -

    Memorize and retrieve the exact radian values for standard angles to ace any quiz on unit circle without hesitation.

  3. Determine point coordinates -

    Calculate the precise (x, y) coordinates for each significant angle, reinforcing unit circle test skills and coordinate mapping.

  4. Apply trigonometric values -

    Use sine, cosine, and tangent values from the unit circle to solve real-world and test-style problems confidently.

  5. Analyze symmetry and patterns -

    Explore the unit circle's quadrantal symmetry and angle patterns to develop effective unit circle memorization game strategies.

Cheat Sheet

  1. Standard Angles & Coordinates -

    Familiarize yourself with the five key angles - 0°, 30°, 45°, 60°, and 90° - and their (cos θ, sin θ) pairs: for example, at 30° the coordinate is (√3/2, 1/2). These values form the backbone of any unit circle quiz or unit circle practice session (see MIT OpenCourseWare). Being able to recall these without hesitation will speed you through any quiz on unit circle fundamentals.

  2. Radian - Degree Conversion -

    Remember the formula radians = degrees × π/180 to switch between systems seamlessly - 90° becomes π/2 and 45° is π/4. A simple mnemonic is "multiply by π, divide by 180" or "π over 180" (Khan Academy). Mastering this conversion is critical for aceing a unit circle test without second-guessing.

  3. Key Sine & Cosine Values at Special Angles -

    Learn that sin 0=0, sin 30=1/2, sin 45=√2/2, sin 60=√3/2, sin 90=1 while cosines run in reverse order (James Stewart, Calculus). Quizzing yourself with fill-in problems on these five pairs cements your recall for any unit circle quiz. Practice mixing and matching sin and cos to build speed and confidence.

  4. Reference Angles & Sign Rules -

    Use the ASTC mnemonic (All Students Take Calculus) to determine where each trig ratio is positive: Quadrant I (all), II (sin), III (tan), IV (cos) per University of Texas resources. First find the reference angle (the acute angle to the x-axis), then apply the sign rule. This technique turns complex quadrant problems into a straightforward two-step process.

  5. Mnemonic Patterns for Rapid Recall -

    Apply the "0-1-2-3-4 finger trick" for sin θ at 0°, 30°, 45°, 60°, 90°: take the square roots of 0,1,2,3,4 then divide by 2 (Purdue University). Cos θ follows the reverse sequence. This unit circle memorization game converts what seems like busywork into a quick, playful pattern you'll never forget.

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