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Quizzes > Mathematics > Advanced Math

Ready to Master Trigonometry? Take the Practice Test Now!

Dive into practical trigonometry problems and ace your trig test!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration with geometric shapes and math symbols on golden yellow background promoting a free trig skills quiz

This trigonometry quiz helps you practice the skills you need on tests: sine, cosine, tangent, and solving triangles, so you can spot gaps before the exam. Get instant feedback with clear steps, then try the extra practice set to focus on the topics you miss.

In a right triangle, the sine of an angle is defined as the ratio of which sides?
Opposite side over hypotenuse
Adjacent side over hypotenuse
Hypotenuse over adjacent side
Opposite side over adjacent side
In a right triangle, the sine of an angle is the length of the side opposite the angle divided by the length of the hypotenuse. This is one of the primary trigonometric ratios and is foundational to triangle trigonometry. Understanding this ratio is essential for solving triangles and for using the unit circle definitions.
What is the value of sin(30°)?
?2/2
0
?3/2
1/2
The sine of 30 degrees corresponds to half the length of the hypotenuse in a 30-60-90 triangle, which yields 1/2. This is a well-known special angle in trigonometry. The value can also be seen from the unit circle where the y-coordinate at 30° (?/6 radians) is 1/2.
What is cos(0°)?
Undefined
0
-1
1
On the unit circle, the cosine of an angle is the x-coordinate. At 0°, the point is (1, 0), so the x-coordinate is 1. Therefore, cos(0°) equals 1.
What is tan(45°)?
?3
1
-1
0
Tangent is the ratio of sine to cosine. At 45°, both sine and cosine are ?2/2, so their ratio is 1. This makes tan(45°) equal to 1.
What is sin(-90°)?
-1
1
Undefined
0
The sine function is odd, meaning sin(-?) = -sin(?). Since sin(90°) = 1, sin(-90°) must be -1. On the unit circle, the point at -90° has a y-coordinate of -1.
Which of the following is the Pythagorean identity?
tan²(x) + 1 = sin²(x)
sin²(x) + cos²(x) = 1
1 + cot²(x) = csc²(x)
sec²(x) + tan²(x) = 1
The fundamental Pythagorean identity in trigonometry is sin²(x) + cos²(x) = 1 for all real x. It is derived from the Pythagorean theorem applied to the unit circle. This identity underpins many other trigonometric relationships and simplifications.
Which trigonometric function is the reciprocal of sine?
Cotangent (cot)
Secant (sec)
Tangent (tan)
Cosecant (csc)
The cosecant function, abbreviated csc, is defined as the reciprocal of the sine function: csc(x) = 1/sin(x). Reciprocal identities pair sine with cosecant and cosine with secant, and tangent with cotangent. This is useful when rewriting expressions or solving equations.
On the unit circle, what are the coordinates of the point at 90°?
(0, -1)
(0, 1)
(-1, 0)
(1, 0)
The unit circle has radius 1 centered at the origin. At 90° (?/2 radians), the point lies on the positive y-axis at (0, 1). This coordinate gives cos(90°)=0 and sin(90°)=1.
What are all solutions for x in [0°, 360°] for sin(x) = 0.5?
30°, 150° and 330°
150° and 210°
0° and 180°
30° and 150°
The sine function equals 0.5 at 30° (?/6) in the first quadrant and at 150° (5?/6) in the second quadrant for one full cycle from 0° to 360°. No other angles in that interval produce 0.5. Understanding the unit circle and symmetry helps identify these solutions.
Which equation represents the Law of Sines in a triangle with sides a, b and angles A, B?
sin(A)/a = sin(B)/b
a/sin(A) = b/sin(B)
a/sin(B) = b/sin(A)
a·sin(A) = b·sin(B)
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/sin(A) = b/sin(B) = c/sin(C). This is used to solve for unknown sides or angles in non-right triangles. It is derived from extending heights and using right triangle definitions.
Convert 150° to radians.
5?/6
3?/4
?/3
2?/3
To convert degrees to radians, multiply by ?/180. Thus, 150° × ?/180 = (150/180)? = 5?/6. This conversion is essential for calculus and other advanced applications.
Solve for x in [0°, 180°]: tan(x) = ?3.
30°
150°
120°
60°
Tangent of 60° is ?3. In the interval from 0° to 180°, the tangent function is positive only in the first quadrant (0° to 90°), which gives x = 60°. No other angles in that range have a tangent of ?3.
What is the double-angle formula for sin(2?)?
1 - 2 sin²(?)
2 sin(?) cos(?)
tan(?) + cot(?)
sin²(?) - cos²(?)
The double-angle identity for sine is sin(2?) = 2 sin(?) cos(?). This is derived from the sum formula sin(a+b) = sin(a)cos(b) + cos(a)sin(b) by setting a = b = ?. It is frequently used to simplify expressions and solve equations.
What is the period of the function y = 2 sin(3x)?
?/3
6?
2?/3
2?
The period of sin(bx) is 2?/b. Here b = 3, so the period is 2?/3. The amplitude of 2 does not affect the period, it only scales the vertical stretch. Understanding period adjustment is key to graphing trigonometric functions.
In the function y = 3 cos(2x), what is the amplitude?
2
6
1
3
In y = a cos(bx + c) + d, the amplitude is |a|. Here a = 3, so the amplitude is 3. It represents the maximum distance from the midline of the graph. The coefficient 2 affects the period, not the amplitude.
Simplify the expression sin²(x) - cos²(x).
-cos(2x)
sin(2x)
1 - 2 cos²(x)
cos(2x)
Using the cosine double-angle identity cos(2x) = cos²(x) - sin²(x), we can rearrange to get sin²(x) - cos²(x) = -cos(2x). Recognizing and applying these identities simplifies many expressions.
In triangle ABC, with sides a = 7, b = 10 and angle C = 60°, what is the length of side c using the Law of Cosines?
?79
?119
13
79
The Law of Cosines states c² = a² + b² - 2ab cos(C). Substituting yields c² = 49 + 100 - 2·7·10·0.5 = 149 - 70 = 79, so c = ?79. This formula generalizes the Pythagorean theorem to all triangles.
Which of the following expressions is equivalent to tan(x) + cot(x)?
2 sin(x) cos(x)
1 / (sin(x) cos(x))
sin(2x)
cos(2x)
Express tan(x) as sin(x)/cos(x) and cot(x) as cos(x)/sin(x), then combine over a common denominator: [sin²(x) + cos²(x)]/[sin(x) cos(x)] = 1/[sin(x) cos(x)] by the Pythagorean identity. This simplification is common in advanced trigonometric manipulations.
What is arccos(1/2) in radians?
?/6
?/2
2?/3
?/3
Arccosine gives the angle whose cosine is the given value. Since cos(?/3) = 1/2, arccos(1/2) = ?/3. It lies within the principal value range [0, ?].
Solve for x in [0, 2?): sin(x) + cos(x) = 1.
?/4 and 3?/4
?/6 and 5?/6
?/2 and ?
0 and ?/2
Rewrite sin(x)+cos(x) = ?2 sin(x + ?/4). Setting ?2 sin(x + ?/4) = 1 gives sin(x + ?/4) = 1/?2, so x + ?/4 = ?/4 or 3?/4. Thus x = 0 or ?/2 within [0,2?).
Which is the half-angle formula for sin(?/2)?
±?((1 - cos(?)) / 2)
?((1 + sin(?)) / 2)
1 - 2 sin²(?/2)
±?((1 + cos(?)) / 2)
The half-angle identity for sine is sin(?/2) = ±?[(1 - cos(?))/2], where the sign depends on the quadrant of ?/2. This formula comes from the cosine double-angle identity. It is useful for integration and solving equations involving half angles.
What formula gives the area of a triangle in terms of two sides and the included angle?
ab sin(C)
½ ab sin(C)
½ ab cos(C)
ab cos(C)
The area of a triangle can be computed as (1/2)ab sin(C), where a and b are two sides and C is the included angle. This derives from dropping a height and using the definition of sine. It generalizes the base-height formula to non-right triangles.
Solve for x in [0°, 360°] if sec(x) = -2.
90° and 270°
120° and 240°
60° and 300°
180° only
Secant is the reciprocal of cosine, so cos(x) = -1/2. This occurs at 120° and 240° in the [0°, 360°] interval. Identifying cosine values and their corresponding angles is crucial for solving such equations.
What is the formula for tan(A + B) in terms of tan(A) and tan(B)?
tan(A) tan(B) - 1
(tan(A) + tan(B)) / (1 - tan(A) tan(B))
tan(A) + tan(B)
(tan(A) + tan(B)) / (1 + tan(A) tan(B))
The sum formula for tangent is tan(A + B) = [tan(A) + tan(B)]/[1 - tan(A) tan(B)]. This is derived from dividing the sine and cosine sum formulas. It is essential for advanced angle addition and transformation problems.
Which identity expresses sin(3?) in terms of sin(?)?
sin(?) + sin(2?)
3 sin(?) - 4 sin³(?)
3 sin²(?) - 1
4 sin³(?) - 3 sin(?)
The triple-angle identity for sine is sin(3?) = 3 sin(?) - 4 sin³(?). It is derived using sum formulas repeatedly or binomial expansion with Euler's formula. This identity simplifies expressions involving sin(3?) in terms of the basic sine function.
0
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Study Outcomes

  1. Apply primary trigonometric ratios -

    Use sine, cosine, and tangent to compute unknown angles and sides in right triangles confidently.

  2. Solve non-right triangle challenges -

    Employ the Law of Sines and the Law of Cosines to tackle practical problems involving oblique triangles.

  3. Analyze unit circle concepts -

    Interpret trigonometric values on the unit circle to connect algebraic functions with geometric angle measures.

  4. Evaluate solutions with instant feedback -

    Identify misconceptions and reinforce correct problem-solving methods by reviewing real-time results.

  5. Implement strategic exam techniques -

    Manage time and accuracy effectively to boost your performance on trig tests and trigonometry exams.

  6. Build confidence with varied practice questions -

    Strengthen your skills through a range of problems, from basic computations to complex word scenarios.

Cheat Sheet

  1. Fundamental Trig Ratios -

    Master the core definitions of sine, cosine, and tangent in right triangles - sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use the mnemonic "SOH-CAH-TOA" to recall these ratios quickly when tackling your trigonometry test. Reinforcing these basics from sources like Khan Academy ensures you're solid on every trig test question.

  2. Unit Circle and Radian Measure -

    Familiarize yourself with the unit circle's key angles (0, 30°, 45°, 60°, 90°) and their radian equivalents (0, π/6, π/4, π/3, π/2) along with coordinate values like (√2/2, √2/2). This foundation, advised by MIT OpenCourseWare, helps you convert between degrees and radians instantly during a trigonometry exam. Being fluent in these values will speed you through any trigonometry practice questions.

  3. Essential Trig Identities -

    Review identities like sin²θ + cos²θ = 1, the double-angle formulas (e.g., sin 2θ = 2 sin θ cos θ), and angle-sum identities (sin(A±B) = sin A cos B ± cos A sin B). According to Stewart's Calculus, these allow you to simplify complex expressions and verify answers on your trig test. Regular practice with these identities ensures you can recall them under exam pressure.

  4. Solving Trigonometric Equations -

    Learn to solve equations like sin θ = 1/2 by finding principal angles (π/6, 5π/6) and then general solutions θ = π/6 + 2πn or 5π/6 + 2πn, where n ∈ ℤ. Leveraging techniques from university-level textbooks will help you handle both single- and multi-step problems in a practical trigonometry problems set. Clear strategies here will boost your confidence on any trig test section involving equation solving.

  5. Real-World Applications -

    Apply trigonometry to measure heights, distances, and angles of elevation/depression - like finding a tree's height using tan θ = height/distance. Practice these scenarios from sources such as the U.S. Department of Education's math resources to nail word problems on your trigonometry exam. Working through varied trigonometry practice questions builds the problem-solving agility needed for real-life and exam success.

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