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Quizzes > High School Quizzes > Mathematics

Ace Your COS Practice Quiz

Sharpen Your Skills with Interactive Practice Questions

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Colorful paper art promoting The Cos Quiz Challenge for high school math students.

Easy
What is the value of cos 0°?
1
0
-1
Undefined
At 0°, the point on the unit circle is (1, 0), so the cosine, which represents the x-coordinate, is 1. Hence, cos 0° equals 1.
What is the value of cos 90°?
0
1
-1
Undefined
At 90°, the unit circle coordinate is (0, 1), so the cosine (x-coordinate) is 0. Therefore, cos 90° equals 0.
How is the cosine of an angle defined in the context of the unit circle?
It is the x-coordinate of the point on the unit circle.
It is the y-coordinate of the point on the unit circle.
It is the distance from the origin to the point.
It is the slope of the line from the origin to the point.
In the unit circle, the cosine corresponds to the x-coordinate of a point formed by an angle. This definition links trigonometric functions to geometry.
Which property best describes the cosine function?
Cosine is an even function (cos(-θ) = cos(θ)).
Cosine is an odd function (cos(-θ) = -cos(θ)).
Cosine has no symmetry.
Cosine is periodic with no specific symmetry.
The cosine function satisfies the property cos(-θ) = cos(θ), which means it is even. This symmetry is fundamental in trigonometry.
What is the approximate value of cos 60°?
0.5
0.707
0.866
1
Cos 60° equals 0.5 exactly, as determined from the standard trigonometric ratios. This value is commonly used in trigonometry problems.
Medium
What is the period of the basic cosine function, y = cos x?
2π
π
4π
1
The cosine function completes one full cycle over an interval of 2π. This periodic behavior is a central property of trigonometric functions.
Identify the amplitude of the function y = 3 cos x.
3
1
2
6
The amplitude of a cosine function is the absolute value of its coefficient, which in this case is 3. Thus, the amplitude is 3.
Determine the phase shift of the function y = cos(x - π/3).
π/3 units to the right
π/3 units to the left
No phase shift
π/6 units to the right
For the function cos(x - π/3), the graph is shifted π/3 units to the right. Phase shift is determined by the expression within the function.
Which identity correctly represents the double-angle formula for cosine?
cos(2θ) = 2cos²θ - 1
cos(2θ) = 1 - 2cos²θ
cos(2θ) = 2sin²θ - 1
cos(2θ) = sin²θ - cos²θ
The identity cos(2θ) = 2cos²θ - 1 is a standard form of the double-angle formula for cosine. It is widely used in trigonometric transformations.
Solve for x in the interval [0, 2π): cos x = 1.
x = 0
x = π
x = π/2
x = 2π
Within the interval [0, 2π), cos x equals 1 only at x = 0. Though cos 2π is also 1, the interval typically excludes 2π because it is identical to 0.
What effect does adding 2 to the cosine function in y = cos x have on its graph?
It shifts the graph vertically upward by 2 units.
It shifts the graph vertically downward by 2 units.
It shifts the graph horizontally to the right by 2 units.
It stretches the graph vertically by a factor of 2.
Adding a constant to a function results in a vertical translation. In this case, adding 2 shifts the cosine graph upward by 2 units.
Which equation represents the law of cosines for a triangle with sides a, b, and c, where angle C is opposite side c?
c² = a² + b² - 2ab cos C
c² = a² + b² + 2ab cos C
c² = a² - b² - 2ab cos C
c² = a² + b² - 2a cos C
The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles, given by c² = a² + b² - 2ab cos C. This formula is essential in solving triangles when trigonometric functions are involved.
Express cos²θ in terms of the cosine double-angle.
cos²θ = (1 + cos(2θ))/2
cos²θ = (1 - cos(2θ))/2
cos²θ = cos(2θ)/2
cos²θ = 2cos(2θ) - 1
The half-angle identity for cosine squared expresses cos²θ as (1 + cos(2θ))/2. This identity is derived from the double-angle formulas and is useful in integration and simplification.
What is the value of cos 180°?
-1
1
0
0.5
At 180°, the point on the unit circle is (-1, 0), so the cosine, which represents the x-coordinate, is -1. This is a standard trigonometric value.
For the function y = 2 cos(3x), what is the period of the cosine function?
2π/3
2π
3(2π)
6π
The period of a cosine function in the form cos(bx) is calculated as 2π/|b|. Here, b = 3, so the period is 2π/3.
Hard
Solve the equation 2 cos² x - 3 cos x + 1 = 0 for x in the interval [0, 2π).
x = 0, π/3, and 5π/3
x = π/3 and 5π/3
x = 0 and π
x = π/2 and 3π/2
Substituting u = cos x, the equation becomes 2u² - 3u + 1 = 0, which factors as (2u - 1)(u - 1) = 0. This gives the solutions cos x = 1/2 (x = π/3, 5π/3) and cos x = 1 (x = 0) within the interval [0, 2π).
Given that cos A = 0.6 with A in the first quadrant, what is the value of sin A?
0.8
0.6
0.4
1.0
Using the identity sin²A + cos²A = 1, we find sin²A = 1 - 0.36 = 0.64, hence sin A = 0.8. Since A is in the first quadrant, the sine is positive.
What is the smallest positive period T such that cos(x + T) = cos x for all x?
2π
π
π/2
4π
The cosine function is periodic with a period of 2π, meaning that cos(x + 2π) = cos x for every x. This is the smallest positive period for the cosine function.
Determine the exact value of cos(7π/6).
-√3/2
√3/2
-1/2
1/2
The angle 7π/6 is in the third quadrant where cosine is negative, and its reference angle is π/6. Since cos(π/6) is √3/2, we get cos(7π/6) = -√3/2.
Express the cosine of the sum of two angles using an identity.
cosα cosβ - sinα sinβ
cosα sinβ + sinα cosβ
cosα cosβ + sinα sinβ
sinα cosβ - sinβ cosα
The cosine sum formula is given by cos(α + β) = cosα cosβ - sinα sinβ. This identity is fundamental for manipulating trigonometric expressions involving sums of angles.
0
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Study Outcomes

  1. Understand the properties and behavior of the cosine function.
  2. Apply trigonometric identities to solve cosine-related problems.
  3. Analyze transformations of cosine graphs to interpret shifts and stretches.
  4. Evaluate cosine values for given angles within right-angled triangles.
  5. Solve real-world problems using cosine concepts and models.

Cos Quiz: Practice Test Cheat Sheet

  1. Definition of Cosine - In a right triangle, cosine is the ratio of the length of the adjacent side to the hypotenuse. It's part of the classic SOH‑CAH‑TOA mnemonic, making it easy to recall. Keep this definition at your fingertips whenever a triangle pops up! Trigonometry Mnemonics
  2. Key Angle Values - Commit to memory the cosine values for the big five angles: cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2, cos(90°)=0. These are your go‑to tools for speedy problem solving and mental checks. Practice shooting them back whenever you see a special angle. Special Angles Cosine Values
  3. Pythagorean Identity - The identity sin²(θ)+cos²(θ)=1 links sine and cosine in a beautiful balance. It's a powerhouse for simplifying expressions and rearranging equations on the fly. Whenever you're stuck, this little formula will save the day. Pythagorean Identities
  4. Unit Circle Cosine - On the unit circle, cosine tells you the x‑coordinate of a point at angle θ. Visualizing it this way reveals why cosine repeats every 2π and swings between - 1 and 1. Spin that circle in your mind to see periodicity in action! Cosine on the Unit Circle
  5. Signs in Quadrants - Cosine is positive in Quadrants I and IV, but flips negative in Quadrants II and III. Remember "A Smart Trig Class" to map out which functions shine where. A quick quadrant sketch can clear sign confusion instantly. Cosine Sign Chart
  6. Addition & Subtraction Formulas - Master cos(A±B)=cosA·cosB∓sinA·sinB to handle sums and differences of angles. These formulas break down complex angles into familiar pieces so you can compute tricky values by hand. Drill them until you can recall them without a second thought! Addition & Subtraction Formulas
  7. Double‑Angle Formula - The formula cos(2θ)=cos²(θ)−sin²(θ) is perfect for tackling problems with angles doubled or halved. It often simplifies messy expressions or solves equations in a snap. Play around with it, and watch how it transforms your trigonometry toolkit. Double‑Angle Formula
  8. Reciprocal Relationship - Secant is just the reciprocal of cosine: sec(θ)=1/cos(θ). Knowing this link makes switching between secant and cosine smooth: no more guesswork when solving secant equations! Keep this swap in mind when you see a sec(θ) pop up. Secant Relationship
  9. Law of Cosines - For any triangle, c²=a²+b²−2ab·cos(C) extends Pythagoras so you can handle non‑right angles. It's a must‑know when side lengths and an angle are given. Think of it as Pythagorean's cool cousin who visits whenever you need that extra cosine power! Law of Cosines
  10. Graphing Cosine - The cosine graph starts at (0,1), dips down to (π, - 1), and bounces back every 2π. Visualizing its smooth wave helps you anticipate shifts, stretches, and reflections. Sketch it out, label the peaks and valleys, and make friends with its rhythm! Cosine Function Graph
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