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Quizzes > Quizzes for Business > Education

Unit Circle Trigonometry Quiz Challenge

Sharpen Your Trigonometry Angle and Radian Skills

Difficulty: Moderate
Questions: 20
Learning OutcomesStudy Material
Colorful paper art promoting a Unit Circle Trigonometry Quiz

Ready to master the unit circle? This unit circle trigonometry quiz guides you through angle positions, radian conversions, and sine and cosine values for all four quadrants. Perfect for students or educators aiming to strengthen trigonometry fundamentals, it offers instant feedback and can be freely customized in our editor. For extra practice, check out Circle Geometry Quiz Class 9 or the Knowledge Assessment Quiz . Discover more quizzes to enhance your math skills today!

What is sin(0°)?
 
-1
0
1
The sine of 0° corresponds to the y-coordinate on the unit circle at 0°, which is 0. Sine measures vertical displacement from the origin.
What is cos(90°)?
-1
 
0
1
Cosine at 90° gives the x-coordinate on the unit circle, which is 0. Cosine measures horizontal displacement from the origin at that angle.
What is sin(90°)?
1
 
0
-1
Sine of 90° equals the y-coordinate at the top of the unit circle, which is 1. This reflects maximum vertical displacement.
What is cos(0°)?
1
-1
0
 
Cosine at 0° gives the x-coordinate of the unit circle at the positive x-axis, which is 1. This represents maximum horizontal displacement.
What is cos(180°)?
1
0
-1
 
Cosine of 180° corresponds to the point on the negative x-axis of the unit circle, which has x-coordinate -1. This indicates full negative horizontal displacement.
What are the coordinates of the point on the unit circle at 45°?
(√2/2, √2/2)
(√3/2, 1/2)
(1/2, √3/2)
(-√2/2, √2/2)
At 45°, both x and y coordinates equal √2/2 on the unit circle. This reflects the 45° isosceles right triangle with equal legs.
What is sin(π/6)?
-1/2
1/2
√2/2
√3/2
π/6 corresponds to 30°, where the y-coordinate on the unit circle is 1/2. Sine of 30° is 1/2 as a fundamental reference value.
What is cos(π/3)?
1/2
-1/2
√2/2
√3/2
π/3 equals 60°, where the x-coordinate on the unit circle is 1/2. Cosine of 60° is 1/2 from a standard right triangle.
What is tan(π/4)?
1
0
-1
 
π/4 equals 45°, where sine and cosine are equal (√2/2), making their ratio tan = 1. This is a key property of the 45° angle.
What are the coordinates of the point on the unit circle at 225°?
(√2/2, -√2/2)
(-√3/2, -1/2)
(-1/2, -√3/2)
(-√2/2, -√2/2)
225° is 180°+45°, so the coordinates are both negative and equal to -√2/2. This reflects symmetry in the third quadrant.
In which quadrant are cosine values negative and sine values positive?
II
III
I
IV
In the second quadrant, x (cosine) is negative and y (sine) is positive. This quadrant covers angles between 90° and 180°.
Convert 2π/3 radians to degrees.
60°
240°
90°
120°
Multiplying 2π/3 by 180/π gives 120°. This is a standard radian-to-degree conversion.
What is the reference angle for 5π/4?
π/4
3π/4
π/2
π/3
5π/4 is in the third quadrant and its distance from π is π/4. The reference angle is always the acute angle to the x-axis.
What is sin(3π/2)?
1
-1
-√2/2
0
3π/2 corresponds to 270°, where the point on the unit circle is (0, -1). The y-coordinate gives sine = -1.
What is sec(π/4)?
√2
1
2
1/√2
Secant is the reciprocal of cosine. Cosine of π/4 is √2/2, so its reciprocal sec(π/4) = 2/√2 = √2.
If sin(θ) = √3/2 and θ is in the second quadrant, what is θ?
2π/3
π/3
4π/3
5π/3
√3/2 corresponds to 60° (π/3). In the second quadrant, the angle is π - π/3 = 2π/3. This satisfies sine symmetry.
Find θ in (0,2π) if tan(θ) = √3 and cos(θ) < 0.
4π/3
π/3
2π/3
5π/3
Tangent √3 has reference angle π/3. For tan positive and cos negative, θ must be in the third quadrant: π + π/3 = 4π/3.
What is sin(-π/6)?
-1/2
√3/2
1/2
-√3/2
Sine is an odd function: sin(-x) = -sin(x). Since sin(π/6) = 1/2, sin(-π/6) = -1/2.
Evaluate cos(3π/2 + π/4).
-√2
√2/2
√2
-√2/2
Using addition: cos(A+B)=cosAcosB - sinA sinB. cos(3π/2)=0, sin(3π/2)=-1 so value is 0·√2/2 - (-1·√2/2)=√2/2.
What is the radian measure of the angle with reference angle π/4 in the third quadrant?
3π/4
5π/4
7π/4
π/4
In the third quadrant, the angle is π + π/4 = 5π/4. Reference angles add to π in QII and QIII accordingly.
0
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Learning Outcomes

  1. Master sine and cosine values at key unit circle angles.
  2. Identify coordinates for angles in all four quadrants.
  3. Analyze relationships between reference angles and trigonometric values.
  4. Apply radian and degree conversions effectively.
  5. Demonstrate understanding of trigonometric function symmetry.

Cheat Sheet

  1. Understand the Unit Circle - The unit circle is a radius-1 circle centered at the origin that turns angles into coordinate pairs, forming the backbone of trigonometry. By viewing sine as the y-coordinate and cosine as the x-coordinate, you can visually connect angles to function values. Explore the Unit Circle
    2. Math Is Fun
  2. Memorize Key Angle Values - Angles like 30°, 45°, and 60° show up everywhere, and knowing their sine and cosine by heart helps you breeze through problems. For instance, sin(30°)=1/2, cos(30°)=√3/2, while both sin(45°) and cos(45°) equal √2/2. Angle Value Reference
    3. Math.net
  3. Learn the Pythagorean Identity - Derived straight from the Pythagorean theorem, the identity sin²(θ) + cos²(θ) = 1 keeps your trig expressions neatly in check. It lets you swap and simplify between sine and cosine, smoothing out equation solving. Deep Dive
    4. Wikipedia
  4. Understand Quadrant Sign Rules - The unit circle splits into four quadrants, each telling you whether sine and cosine are positive or negative in that region. Use the playful CAST rule to remember: Cosine in the Fourth, All in the First, Sine in the Second, Tangent in the Third. Quadrant Cheat Sheet
    5. IntoMath
  5. Convert Between Degrees and Radians - Degrees and radians are just two ways to measure angles, like miles and kilometers for distance. Multiply degrees by π/180 to get radians (180° = π), or multiply by 180/π to turn radians back to degrees. Get Converting
    6. Math Is Fun
  6. Use Reference Angles - A reference angle is the smallest angle between your terminal side and the x-axis, giving you a shortcut to find trig values in any quadrant. Just find the acute counterpart, check the sign, and plug into your known sine or cosine. Reference Angle Guide
    7. Math.net
  7. Recognize Symmetry in the Unit Circle - The unit circle mirrors itself, meaning some trig values repeat or flip signs as you cross axes. For example, sin(θ) = sin(180° − θ) and cos(θ) = −cos(180° − θ), a pattern you can use to cut your work in half. Symmetry Tricks
    8. SparkNotes
  8. Practice with Special Triangles - The 30°-60°-90° and 45°-45°-90° triangles are the secret sauce for unit circle values, with known side ratios that plug right into sine and cosine. A 45°-45°-90° triangle has legs of 1 and hypotenuse √2, while the 30°-60°-90° triangle uses ratios 1:√3:2. Triangle Breakdown
    9. Owlcation
  9. Understand the Relationship Between Sine and Cosine - Sine and cosine are co-functions, meaning they swap when you shift an angle by 90°: sin(90° − θ) = cos(θ) and cos(90° − θ) = sin(θ). This fun fact can turn tough problems into quick one-liners. Cofunction Practice
    10. Pearson
  10. Apply the Unit Circle to Real-World Problems - Beyond homework, the unit circle describes waves, tides, seasons, and any periodic event, turning abstract numbers into real-life patterns. See how sine waves pop up in music, biology, and engineering to appreciate trig's everyday magic. Real-World Uses
    11. IntoMath
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