Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > Mathematics

Trigonometry Unit Test: Trigonometric Identities (Version B)

Quick, free trigonometry practice quiz on identities. Instant results.

Editorial: Review CompletedCreated By: Leah DwyerUpdated Aug 24, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
paper art illustration of trigonometry quiz triangles protractor and real world problems on dark blue background

This trigonometry unit test (Version B) helps you practice and verify trigonometric identities across angles, functions, and real-world problems. Work through mixed questions at your pace, then review with verifying trig identities, brush up on key points in a unit circle quiz, or try a quick trig identity quiz for extra practice.

In a right triangle, the sine of an acute angle is defined as the ratio of which sides?
Adjacent over Hypotenuse
Adjacent over Opposite
Opposite over Hypotenuse
Opposite over Adjacent
By SOHCAHTOA, the sine of an angle in a right triangle equals the length of the opposite side divided by the hypotenuse. This fundamental definition is used to relate angles to side lengths in trigonometry. Understanding this ratio is key to solving many right triangle problems.
Convert 30° to radians.
?/2
?/3
?/6
?/4
To convert degrees to radians, multiply by ?/180. Thus 30°×(?/180)=?/6. This conversion is essential when working with trigonometric functions in calculus and advanced applications.
On the unit circle, what is the value of cosine at 90°?
1
0
?1
?2/2
On the unit circle, the x-coordinate corresponds to the cosine value. At 90°, the point is (0,1), so cosine(90°)=0. This is a fundamental value used in many trigonometric calculations and proofs.
If sin? = 0.5 for an acute angle ?, what is ? in degrees?
45°
60°
30°
90°
The sine of 30° equals 0.5, and because the angle is acute, ? must be 30°. Recognizing common angle values for sine, cosine, and tangent is crucial for solving problems quickly.
What is the value of tan45°?
1
0
?2
?3
Tangent is the ratio of sine to cosine. For 45°, both sine and cosine are ?2/2, so tan45° = (?2/2)/(?2/2) = 1. This is a key ratio in trigonometry.
What is cos(?/2) exactly?
0
1
?1
?2/2
Cosine of ?/2 radians (90°) corresponds to the x-coordinate on the unit circle at that angle, which is zero. This exact value is foundational in trigonometric identities and unit circle analysis.
In which quadrant is sine positive and cosine negative?
III
I
IV
II
In the second quadrant, x-coordinates (cosine) are negative and y-coordinates (sine) are positive on the unit circle. Identifying the signs of trig functions in quadrants is critical for solving equations.
If sinA = 3/5 and A is acute, what is cosA?
?6/3
4/5
?11/5
2/3
Using the Pythagorean identity, cosA = ?(1?sin²A) = ?(1?(3/5)²) = ?(1?9/25) = ?(16/25) = 4/5. This applies since A is acute, making cosine positive.
Which of the following is the formula for the law of sines in any triangle?
sinA/a = sinB/b
a/sinB = c/sinA
(a + b)/sinC
a/sinA = b/sinB
The law of sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sinA = b/sinB = c/sinC. This is used for solving oblique triangles.
What is the exact radian measure for 150°?
2?/3
4?/5
5?/6
3?/4
To convert 150° to radians, multiply by ?/180: 150×(?/180) = 5?/6. Exact radian measures are important for higher-level trigonometry and calculus.
Which angle(s) between 0 and 2? satisfy sin? = ?3/2?
?/6 and 5?/6
5?/6 and 7?/6
2?/3 and 5?/3
?/3 and 2?/3
The equation sin? = ?3/2 holds for angles where the y-coordinate on the unit circle is ?3/2: ? = ?/3 and 2?/3 in [0,2?). Recognizing these values helps solve trigonometric equations.
A triangle has sides a = 8, b = 6, and included angle C = 60°. What is its area?
12?3
6?3
24?3
14?3
The area of a triangle with SAS data is (1/2)ab·sinC = 0.5×8×6×sin60° = 24×(?3/2) = 12?3. This formula generalizes the Pythagorean area for non-right triangles.
According to the Pythagorean identity, what is sin²? + cos²? equal to?
2
sin2?
0
1
The Pythagorean identity states that for all angles ?, sin²? + cos²? = 1. This fundamental identity underpins many trigonometric simplifications.
What is the exact value of cos75°?
(?3 ? 1)/(2?2)
(?6 ? ?2)/4
(?3 + 1)/(2?2)
(?6 + ?2)/4
Using the sum of angles formula, cos75° = cos(45°+30°) = cos45°cos30° ? sin45°sin30° = (?2/2)(?3/2) ? (?2/2)(1/2) = (?6 ? ?2)/4. Knowing exact values aids in advanced problem solving.
What is the trigonometric identity for sin(3?)?
sin? + sin2?
sin²? + sin?
3 sin? ? 4 sin³?
4 sin? ? 3 sin³?
The triple-angle identity for sine is sin(3?) = 3 sin? ? 4 sin³?. This identity is derived from angle-sum formulas and is useful in solving higher-order equations.
A 10-meter ladder leans against a wall, forming a 75° angle with the ground. Approximately how high does the ladder reach on the wall?
2.59 m
9.66 m
8.66 m
5.00 m
The height reached is the opposite side: 10·sin75° ? 10×0.9659 = 9.659 m. This real-world application of sine relates angles to heights in right triangles.
Given triangle ABC where A = 30°, a = 10, and b = 15, how many distinct triangles satisfy these conditions?
3
1
0
2
Using the law of sines, sinB = b·sinA/a = 15·0.5/10 = 0.75, giving two possible angle B solutions (acute and obtuse) and thus two triangles (the ambiguous case).
Simplify the expression sec? ? cos?.
sin²?
sec? cos?
1
tan? sin?
sec? ? cos? = (1/cos?) ? cos? = (1?cos²?)/cos? = sin²?/cos? = sin?·(sin?/cos?) = sin?·tan?. Simplifying with Pythagorean identities reveals this result.
Which identity represents the sum-to-product for sinA + sinB?
sinA + sinB = sin(A + B)
sinA cosB + cosA sinB
2 cos((A + B)/2) sin((A + B)/2)
2 sin((A + B)/2) cos((A ? B)/2)
The sum-to-product identity for sine is sinA + sinB = 2 sin((A + B)/2) cos((A ? B)/2). This transforms sums of sines into products, simplifying integration and solving equations.
What is the law of cosines for a triangle with sides a, b, and c opposite angles A, B, and C respectively?
c = a + b ? 2ab cosC
c² = a² + b² + 2ab cosC
c² = a² + b² ? 2ab cosC
c² = a² ? b² ? 2ab cosC
The law of cosines generalizes the Pythagorean theorem: c² = a² + b² ? 2ab cosC, relating all three sides and the included angle. It's essential for solving oblique triangles when given SAS.
What is the exact value of tan(3?/4)?
1
?3
??3
?1
3?/4 radians corresponds to 135°, where sine and cosine are (?2/2) and (??2/2) respectively. Therefore tan(3?/4) = (?2/2)/(??2/2) = ?1. Knowing signs in quadrants is key.
Which of the following gives the solutions of 4 sin²x ? 4 sin x + 1 = 0 for x in [0, 2?)?
x = 7?/6 and 11?/6
x = ?/3 and 2?/3
x = ?/4 and 3?/4
x = ?/6 and 5?/6
The quadratic factors as (2 sin x ? 1)² = 0, so sin x = 1/2. On [0,2?), this yields x = ?/6 and 5?/6. Advanced trig equations often require quadratic techniques.
Which identity correctly expresses cos³? in terms of cos? and cos3??
(1/4)(cos? ? cos3?)
(1/4)(3 cos? + cos3?)
(1/3)(cos? + cos3?)
(3 cos? ? cos3?)/4
Using triple-angle derivations, cos3? = 4 cos³? ? 3 cos?. Solving for cos³? gives (cos3? + 3 cos?)/4 = (1/4)(3 cos? + cos3?). These identities simplify higher powers.
What is the exact value of sin(?/8)?
(?6 ? ?2)/4
(?6 + ?2)/4
?(2 + ?2)/2
?(2 ? ?2)/2
Using the half-angle formula, sin(?/8) = sin(22.5°) = ?((1 ? cos45°)/2) = ?((1 ? ?2/2)/2) = ?(2 ? ?2)/2. Exact half-angle values are common in advanced trigonometry.
0
{"name":"In a right triangle, the sine of an acute angle is defined as the ratio of which sides?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"In a right triangle, the sine of an acute angle is defined as the ratio of which sides?, Convert 30° to radians., On the unit circle, what is the value of cosine at 90°?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}

Study Outcomes

  1. Compute Trigonometric Ratios -

    Calculate sine, cosine, and tangent values for acute and obtuse angles with accuracy, reinforcing foundational trigonometry skills.

  2. Solve Right”Triangle Problems -

    Apply trigonometric ratios to determine unknown side lengths and angles in right triangles using given measurements.

  3. Interpret Real”World Scenarios -

    Translate practical situations into trigonometric problems and use appropriate methods to find solutions.

  4. Apply Strategies from Previous Tests -

    Use techniques from unit 5 trigonometry unit test a answers and unit 7 trigonometry test answers to tackle similar quiz questions effectively.

  5. Analyze Solution Methods -

    Compare different approaches to solving trigonometry test questions and identify the most efficient techniques.

  6. Enhance Exam Readiness -

    Boost your confidence by practicing a variety of trigonometry test questions, ensuring you're prepared for any trigonometry unit test.

Cheat Sheet

  1. Unit Circle Fundamentals -

    Review the unit circle definitions of angles in radians and degrees to master sine, cosine, and tangent values at key points (0, π/6, π/4, π/3, π/2). According to MIT OpenCourseWare, memorizing coordinates like (√2/2, √2/2) for 45° helps you quickly evaluate expressions on the trigonometry unit test version b. Use the "All Students Take Calculus" mnemonic to recall which functions are positive in each quadrant.

  2. SOHCAHTOA and Basic Ratios -

    Internalize the SOHCAHTOA mnemonic to remember that sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. Khan Academy emphasizes practicing varied trigonometry practice quiz questions to solidify these ratios under time pressure. When reviewing unit 5 trigonometry unit test a answers, you'll notice SOHCAHTOA emerges in almost every right”triangle problem.

  3. Pythagorean Identities -

    Master the identity sin²θ + cos²θ = 1 and its derived forms (1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ) by deriving them from the Pythagorean Theorem in a right triangle. Purdue University's online math lab suggests proving each identity with actual triangle sides for deeper understanding. This skill is crucial for both unit 7 trigonometry test answers and challenging equations on the trigonometry unit test version b.

  4. Solving Right Triangles -

    Combine basic trig ratios with the Law of Sines or Cosines (a/sin A = b/sin B = c/sin C) to find missing sides and angles in non”right triangles. The University of California's math department recommends working through diverse trigonometry test questions to become fluent in selecting the right approach. Real exam problems often layer multiple steps - first use SOHCAHTOA, then apply the Pythagorean identity, and finally use inverse functions.

  5. Inverse Trig & Real”World Applications -

    Practice using arcsin, arccos, and arctan to determine angles from known side ratios and apply these in surveying, navigation, or architecture problems, as highlighted by the American Mathematical Society. For example, arctan(height/distance) helps calculate elevation angles in engineering tasks. Consistent practice with these inverse functions boosts confidence for any trigonometry unit test version b scenario.

Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Advanced Math Quizzes

Powered by: Quiz Maker