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Quizzes > High School Quizzes > Mathematics

Trigonometric Practice Quiz: Functions & Inverses

Improve with basic, inverse, and geometry worksheets

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting the Trig Basics Challenge, a high school-level trigonometry quiz.

Easy
What is the definition of the sine function in a right triangle?
Opposite/Hypotenuse
Adjacent/Hypotenuse
Opposite/Adjacent
Hypotenuse/Opposite
The sine function is defined as the ratio of the length of the side opposite the angle to the hypotenuse. This fundamental definition is essential in solving right triangle problems.
What is the range of the sine function?
[-1, 1]
[0, 1]
(-∞, ∞)
[-∞, ∞]
The sine function produces outputs between -1 and 1 for any real angle. This property is a key characteristic used in trigonometric analysis.
Which function is the inverse of the cosine function?
Arccosine
Arcsine
Arctangent
Secant
Arccosine is defined as the inverse of the cosine function, meaning it returns the angle whose cosine equals a given value. It is commonly denoted as cos❻¹ or arccos.
If sin(θ) = 0.5 for an acute angle θ, what is one possible measure of θ?
30°
45°
60°
90°
In a right triangle, sin(30°) equals 0.5, making 30° a valid angle measurement. This fact is one of the most common in basic trigonometry.
What is the domain of the inverse sine function (arcsin)?
Values between -1 and 1 inclusive
All real numbers
Only positive numbers
Values greater than 1
The arcsine function is only defined for input values within the interval [-1, 1]. This restriction mirrors the range of the sine function.
Medium
What is the value of cos(45°)?
0.5
0.7071
1
0
Cos(45°) is equal to √2/2, which approximates to 0.7071. This value is frequently used in trigonometric computations.
Which of the following represents tan(θ) in terms of sine and cosine?
sin(θ)/cos(θ)
cos(θ)/sin(θ)
sin(θ) * cos(θ)
cos(θ) - sin(θ)
Tangent is defined as the ratio of sine to cosine, making sin(θ)/cos(θ) the correct expression. This ratio is a foundational identity in trigonometry.
If cos(θ) = 0.6 for an acute angle, what is the value of sec(θ)?
0.6
1.667
1.2
0.4
Secant is the reciprocal of the cosine function, so sec(θ) = 1/0.6, which is approximately 1.667. This reciprocal relationship is fundamental in trigonometry.
What is the domain of the arctan function?
All real numbers
Only positive numbers
Between -1 and 1
0 to infinity
The arctan function accepts all real numbers as input, making its domain the entire set of real numbers. This is a contrast to the inverse sine and cosine functions which have restricted domains.
Which of the following trigonometric identities is correct?
sin²(θ) + cos²(θ) = 1
1 + cos²(θ) = tan²(θ)
sin(θ) * cos(θ) = 1
cos²(θ) - sin²(θ) = 1
The Pythagorean identity sin²(θ) + cos²(θ) = 1 is valid for all angles and is crucial in trigonometry. The other options do not correctly represent trigonometric relationships.
How do you determine the inverse of a trigonometric function such as sin(x)?
By swapping the input and output values
By squaring the function
By taking the reciprocal of the function
By multiplying the function by -1
Finding an inverse function involves interchanging the roles of the input and output. This method is used to derive inverse trigonometric functions like arcsin.
What is the range of the arccos function?
[0, π]
[-π/2, π/2]
[-π, 0]
[0, 2π]
The arccos function outputs angles in the interval [0, π]. This restriction is necessary to ensure that the inverse function is single-valued.
What is the relationship between the sine of an angle and the cosine of its complement?
sin(θ) = cos(90° - θ)
sin(θ) = 1 - cos(θ)
sin(θ) = cos(θ)
sin(θ) = sin(90° - θ)
The complementary angle identity states that the sine of an angle is equal to the cosine of its complement. This is a pivotal concept in understanding trigonometric relationships.
If tan(θ) = 1 for an acute angle, what is the measure of θ in degrees?
45°
30°
60°
90°
When the tangent of an angle equals 1, it indicates that the sine and cosine are equal, which is true for a 45° angle in a right triangle. This is a standard trigonometric result.
What does the term 'inverse trigonometry' refer to?
Finding an angle given a trigonometric ratio
Finding the reciprocal of a trigonometric function
Inverting the graph of a trigonometric function
Removing restrictions on trigonometric functions
Inverse trigonometry involves determining the angle that corresponds to a particular trigonometric ratio. This process is essential when solving for unknown angles in various problems.
Hard
What is the principal value of sin❻¹(-0.5) in degrees?
-30°
30°
150°
-150°
The principal value of the inverse sine function is defined within the range [-90°, 90°]. Since sin(-30°) equals -0.5, sin❻¹(-0.5) is -30°.
Evaluate tan(sin❻¹(0.6)).
0.75
0.67
1.25
0.6
Let θ = sin❻¹(0.6), which implies sin(θ) = 0.6. Using the identity sin²(θ) + cos²(θ) = 1, we find cos(θ) = 0.8. Therefore, tan(θ) = 0.6/0.8 = 0.75.
Evaluate cos(arcsin(0.8)).
0.6
0.8
0.36
1.0
If θ = arcsin(0.8), then sin(θ) = 0.8. Using the identity sin²(θ) + cos²(θ) = 1, we calculate cos(θ) as √(1 - 0.64) = 0.6. This method is common in evaluating composite trigonometric functions.
What is the sum of sin❻¹(x) and cos❻¹(x) for any x in the interval [-1, 1]?
π/2
π
1
0
A well-known identity in inverse trigonometry states that sin❻¹(x) + cos❻¹(x) always equals π/2 for any x between -1 and 1. This identity is frequently used in proofs and problem solving.
If tan❻¹(2) = θ in radians, which of the following is true about θ?
θ is an acute angle between 0 and π/2
θ is an obtuse angle
θ is a negative angle
θ is exactly π/4
Since tan❻¹(2) returns an angle whose tangent is positive and the inverse tangent function outputs angles in the interval (-π/2, π/2), the angle must be acute and lie between 0 and π/2.
0
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Study Outcomes

  1. Define inverse trigonometric functions and describe their domains and ranges.
  2. Calculate the angle corresponding to a given trigonometric value using inverse functions.
  3. Apply inverse trigonometric concepts to solve equations and verify solutions.
  4. Analyze graphs of inverse trigonometric functions to identify key features.
  5. Synthesize knowledge of trigonometric and inverse trigonometric functions to solve real-world problems.

Trigonometry Practice Quiz & Worksheets Cheat Sheet

  1. Understand inverse trig definitions - Inverse trig functions like arcsine (sin❻¹), arccosine (cos❻¹) and arctangent (tan❻¹) let you reverse-engineer angles from given ratios. They're your secret decoder ring when a triangle throws you a curveball - if sin(θ)=0.5, then θ=sin❻¹(0.5)! Byju's: Inverse Trig Basics
  2. Memorize domains and ranges - Knowing valid inputs and outputs prevents ugly mistakes. For example, sin❻¹(x) only accepts x between - 1 and 1 and spits angles between - π/2 and π/2. It's like staying in the safe lane on a roller coaster - you'll enjoy the ride more! Byju's: Domain & Range
  3. Master key identities - Identities such as sin❻¹( - x)= - sin❻¹(x) and sin❻¹(x)+cos❻¹(x)=π/2 are your best friends for simplifying scary expressions. They turn big problems into bite‑sized snacks and boost your algebraic confidence. Byju's: Trig Formulas
  4. Convert between trig and inverse trig - If θ=sin❻¹(x), then sin(θ)=x. Practicing these swaps makes you faster than a calculator at solving puzzles. It's brain training that pays off on every exam! GeeksforGeeks: Function Properties
  5. Visualize with graphs - Plotting arcsine, arccosine and arctangent shows how they behave and why their ranges are limited. A quick sketch can save hours of confusion when you're under the gun. Byju's: Graphical Insights
  6. Learn the derivatives - Cool rules like d/dx[sin❻¹(x)] = 1/√(1 - x²) help you tackle integrals and differentiation problems involving inverse trig. They're essential weapons in your calculus arsenal. Byju's: Differentiation Rules
  7. Solve inverse trig equations - Practice equations like sin❻¹(x)=π/6 to find x=1/2, so you can breeze through homework without breaking a sweat. Repetition turns confusion into clarity! Byju's: Practice Problems
  8. Explore function composition - Identities such as sin(sin❻¹(x))=x and sin❻¹(sin(θ))=θ (within domains) show how trig and inverse trig undo each other. They're the math equivalent of pairing socks - everything fits perfectly. GeeksforGeeks: Composition
  9. Use sum and difference formulas - Combining angles? tan❻¹(x)+tan❻¹(y)=tan❻¹((x+y)/(1 - xy)) for xy<1 lets you blend two angles into one neat package. It's like making a math smoothie! GeeksforGeeks: Sum & Difference
  10. Apply to real-world problems - From architecture to navigation, practice finding angles in right triangles using inverse trig. Applying concepts to real life cements your understanding and makes math feel magical. Byju's: Real‑World Applications
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