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Quizzes > Mathematics

Unit circle fill in quiz: practice radians and key trig values

Quick unit circle practice quiz with 20 blanks. Instant results.

Editorial: Review CompletedCreated By: Summer TeetersUpdated Aug 23, 2025
Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art depicting a Circle Clue Challenge trivia quiz, ideal for middle school students.

This quiz helps you fill in the unit circle, so you can recall angles, radians, and sine and cosine values fast. You will answer 20 blanks and see which points to study next. For more practice, try our unit circle practice quiz and a radians unit circle quiz for focused review.

Which of the following best defines a unit circle?
A circle with a diameter of 1
A circle with a radius of 1
A circle with an area of 1
A circle with a circumference of 1
The unit circle is defined as the set of all points exactly one unit away from the center, typically the origin. Its normalized radius makes it a fundamental tool in trigonometry.
What is the value of cos(0) on the unit circle?
Undefined
-1
0
1
At 0 radians, the point on the unit circle is (1, 0), so the cosine, which represents the x-coordinate, is 1. This basic fact is one of the cornerstones of trigonometry.
Which angle in radians corresponds to the point (0,1) on the unit circle?
3π/2
π
π/4
π/2
The point (0,1) is located at 90 degrees, which is equivalent to π/2 radians on the unit circle. This is a standard reference position in trigonometry.
What is the sine value of π/2 on the unit circle?
√2/2
-1
1
0
At π/2 radians, the y-coordinate on the unit circle is 1, and sine is defined as the y-coordinate. This makes the sine value equal to 1.
What is the standard equation of a circle with center (0,0) and radius r?
x + y = r^2
x^2 + y^2 = r
x^2 + y^2 = r^2
xy = r^2
The standard equation for a circle centered at the origin is given by x² + y² = r². This equation represents all points (x, y) that are a distance r from the origin.
What is the sine of 30° (π/6 radians) on the unit circle?
0
√2/2
1/2
√3/2
On the unit circle, sin(30°) equals 1/2, a ratio derived from the 30-60-90 right triangle. This value is one of the most commonly referenced trigonometric ratios.
What is the cosine of 45° (π/4 radians) on the unit circle?
1/2
√2/2
√3/2
1
At 45 degrees, both the sine and cosine of the angle are equal to √2/2 due to the symmetry of the 45°-45°-90° triangle. This is a standard trigonometric result.
Convert 180° into radians.
3π/2
π/2
π
2π
Using the conversion factor where 180° equals π radians, 180° converts directly to π radians. This conversion is essential for working between the two measurement systems.
What is the tangent of 45° (π/4 radians) on the unit circle?
1/√2
0
1
-1
Since tan(θ) is defined as sin(θ)/cos(θ) and for 45° both sine and cosine are equal (√2/2), their ratio is 1. This is a foundational trigonometric identity.
What are the coordinates of the point on the unit circle corresponding to θ = π?
(0, 1)
(0, -1)
(1, 0)
(-1, 0)
At an angle of π radians, the point on the unit circle is located at (-1, 0). This reflects the symmetry of the circle along the horizontal axis.
What is the length of an arc in a unit circle subtended by an angle of θ radians?
1/θ
θ
2θ
θ^2
For a unit circle with radius 1, the arc length is calculated as the product of the radius and the central angle in radians, which simplifies to θ. This direct relationship is a key concept in circle geometry.
How many degrees are in π/3 radians?
30°
120°
90°
60°
Since π radians is equivalent to 180°, dividing 180° by 3 gives 60°. Converting between radians and degrees is a vital skill in trigonometry.
What is the coterminal angle of 13π/6 that lies between 0 and 2π?
5π/6
11π/6
7π/6
π/6
By subtracting 2π (or 12π/6) from 13π/6, the coterminal angle within the range 0 to 2π is π/6. This demonstrates the periodic nature of trigonometric angles.
What is the measure in radians of a full circle?
2π
4π
360
π
A full circle measures 360° which is equivalent to 2π radians. This conversion is fundamental when working with angular measurements.
In which quadrant is the point (-√2/2, -√2/2) located on the unit circle?
Quadrant I
Quadrant II
Quadrant IV
Quadrant III
Both coordinates of the point are negative, indicating that it lies in Quadrant III. Recognizing the sign of the coordinates is essential for identifying positions on the unit circle.
Given that (0.6, y) is a point on the unit circle in Quadrant I, what is the value of y?
0.8
1
0.6
0.4
Substitute x = 0.6 into the equation x² + y² = 1 to get y² = 1 - 0.36 = 0.64. Since the point is in Quadrant I, y is positive, so y = 0.8.
Determine the cosine and sine of 150°.
(√3/2, -1/2)
(-√3/2, 1/2)
(√3/2, 1/2)
(-√3/2, -1/2)
150° is in the second quadrant where cosine is negative and sine is positive. With a reference angle of 30°, cosine is -cos(30°) = -√3/2 and sine is sin(30°) = 1/2.
Find the exact value of tan(π/3) using the unit circle.
-1/√3
-√3
1/√3
√3
At π/3, the sine value is √3/2 and the cosine value is 1/2, so the tangent is (√3/2) divided by (1/2) which equals √3. This is a standard trigonometric result.
What is the length of the chord between the points at angles 0 and π/3 on the unit circle?
√3
2
π/3
1
The chord length in a circle is given by the formula 2R·sin(θ/2). For a unit circle (R = 1) and a central angle of π/3, the chord length is 2 sin(π/6) = 2·(1/2) = 1.
Why is a tangent to a circle perpendicular to the radius at the point of tangency?
Because tangents always pass through the center of the circle.
Because a tangent line touches the circle at exactly one point, forcing the minimal distance from the center, which is achieved perpendicularly.
Because a tangent line is parallel to any radius drawn from the center.
Because a tangent cuts the circle into two equal halves.
A core theorem in circle geometry states that a tangent is perpendicular to the radius at the point of tangency. This is because if the line were not perpendicular, it would intersect the circle at more than one point, contradicting the definition of a tangent.
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Study Outcomes

  1. Understand key circle geometry terminology and definitions.
  2. Analyze the relationships between circle components such as radii, diameters, and circumferences.
  3. Apply unit circle concepts to solve fill-in-the-blank problems.
  4. Evaluate how circle puzzles illustrate fundamental geometric principles.
  5. Synthesize learned geometry skills to enhance test readiness.

Unit Circle Fill in the Blank Cheat Sheet

  1. Understand the Unit Circle - The unit circle is your trusty sidekick in trigonometry: a circle of radius 1 centered at the origin. It helps you visualize angles as points and builds a solid foundation for sine and cosine functions.
  2. Learn the Equation x² + y² = 1 - This simple equation describes every point on the unit circle and springs from the Pythagorean theorem. Mastering it ties geometry and algebra into one neat package for trigonometric adventures.
  3. Map Sine and Cosine to Coordinates - On the unit circle, cosine is the x”coordinate and sine is the y”coordinate of any point. This clever mapping turns angles into numerical values and reveals how these functions behave.
  4. Memorize Key Angles in Degrees & Radians - Get comfy with 0° (0), 30° (π/6), 45° (π/4), 60° (π/3) and 90° (π/2). Knowing these like the back of your hand speeds up problem”solving and strengthens your trigonometry intuition.
  5. Master the Pythagorean Identity - sin²θ + cos²θ = 1 flows directly from x² + y² = 1. It's a must”know shortcut that simplifies complex expressions and keeps your calculations on track.
  6. Learn Signs in Each Quadrant - "All Students Take Calculus" helps you remember which trig functions are positive: Quadrant I (all), II (sin), III (tan), IV (cos). This trick saves time when evaluating angles beyond 90°.
  7. Practice Standard Angle Values - Work out sine and cosine for angles like 45° (both √2/2) or 30° (½ and √3/2). Regular drills boost your speed and confidence for quizzes and exams.
  8. Extend to All Real Numbers - The unit circle framework lets you define trigonometric functions for any real angle, not just acute ones. Embrace rotations beyond 360° to conquer advanced topics easily.
  9. Use Reference Angles - Reference angles are the acute angles between an angle's terminal side and the x-axis. They simplify finding trig values in any quadrant by linking back to familiar acute”angle results.
  10. Rely on the Unit Circle Chart - Keep a color”coded chart of sine and cosine values for quick lookup. This visual cheat sheet cements your memory and keeps you zooming through problems without missing a beat.
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