Quiz: Trigonometric Functions — 21 questions

1. What are the initial side, terminal side, and vertex of an angle?

A. The original ray, rotated ray, and rotation point

B. The circle center, original ray, and rotated ray

C. The rotated ray, original ray, and circle center

D. The rotation point, circle center, and original ray

Answer: (A)

Explanation

An angle describes the rotation of a ray about its initial point: the original ray is the initial side, the rotated ray is the terminal side, and the rotation point is the vertex. Confusing the initial and terminal sides reverses the roles of the two rays.

2. A ray rotates clockwise from its initial position to its terminal position. How is the resulting angle classified?

A. As an undefined angle

B. As a right angle

C. As a negative angle

D. As a positive angle

Answer: (C)

Explanation

Clockwise rotation produces a negative angle, while anticlockwise rotation produces a positive angle. The size of the rotation does not change this sign convention.

3. How is one radian defined geometrically?

A. As the central angle subtending an arc equal to the circle's radius

B. As the angle formed by two perpendicular radii

C. As the central angle subtending an arc equal to the diameter

D. As one three-hundred-and-sixtieth of a complete revolution

Answer: (A)

Explanation

One radian is the angle at the center of a circle subtended by an arc whose length equals the radius; using a unit circle makes that arc length one unit. A degree instead measures a fraction of a complete revolution.

4. Which relationship provides the fundamental conversion between degree and radian measures?

A. $180^\circ=\pi\text{ radians}$

B. $90^\circ=\pi\text{ radians}$

C. $360^\circ=\pi\text{ radians}$

D. $180^\circ=2\pi\text{ radians}$

Answer: (A)

Explanation

A straight angle measures $180^\circ$ and also measures $\pi$ radians, giving the fundamental relationship $180^\circ=\pi\text{ radians}$. A full revolution is instead $360^\circ=2\pi$ radians.

5. What is the radian measure of a $72^\circ$ angle?

A. $\frac{5\pi}{4}$

B. $\frac{2\pi}{5}$

C. $\frac{5\pi}{2}$

D. $\frac{2\pi}{3}$

Answer: (B)

Explanation

Converting degrees to radians uses $\frac{\pi}{180}\times\text{degree measure}$, so $72^\circ\times\frac{\pi}{180}=\frac{2\pi}{5}$. The reciprocal factor $\frac{180}{\pi}$ is used when converting radians to degrees.

6. A circular arc has radius $8$ cm and central angle $\frac{\pi}{3}$ radians. What is its length?

A. $24\pi\text{ cm}$

B. $\frac{3\pi}{8}\text{ cm}$

C. $\frac{8}{3\pi}\text{ cm}$

D. $\frac{8\pi}{3}\text{ cm}$

Answer: (D)

Explanation

For an angle measured in radians, arc length is found from $l=r\theta$, giving $l=8\times\frac{\pi}{3}=\frac{8\pi}{3}\text{ cm}$. The expression $\theta=\frac{l}{r}$ instead calculates the angle when arc length and radius are known.

7. How should the distance travelled by the tip of a rotating hand be calculated?

A. Convert the rotation to degrees and multiply it by the radius

B. Treat the rotation angle itself as the travelled distance

C. Convert the rotation to radians and use the arc-length relation

D. Divide the radius by the rotation measured in radians

Answer: (C)

Explanation

The tip follows a circular arc, so its travelled distance is an arc length calculated with $l=r\theta$ after expressing the rotation in radians. The angle describes the rotation, whereas the arc length describes the distance travelled.

8. A point on the unit circle reached by an angle $x$ has coordinates $(a,b)$. Which coordinate represents $\cos x$?

A. The distance from the origin

B. The angle measured from the x-axis

C. The y-coordinate $b$

D. The x-coordinate $a$

Answer: (D)

Explanation

On the unit circle, the cosine of an angle equals the point’s horizontal coordinate, while the sine equals its vertical coordinate. Confusing cosine with the y-coordinate would assign cosine the role of sine.

9. Which identity holds for every real number $x$?

A. $\sin^2x+\cos^2x=1$

B. $1+\tan^2x=\cosec^2x$

C. $\sin x+\cos x=1$

D. $\sin^2x-\cos^2x=1$

Answer: (A)

Explanation

The fundamental Pythagorean identity states that the squares of sine and cosine add to one. The identity involving $1+\tan^2x$ uses secant, not cosecant, and is a different reciprocal identity.

10. What are the values of $\sin x$ and $\cos x$ when $x=\frac{3\pi}{2}$?

A. $\sin x=-1$ and $\cos x=0$

B. $\sin x=0$ and $\cos x=1$

C. $\sin x=0$ and $\cos x=-1$

D. $\sin x=1$ and $\cos x=0$

Answer: (A)

Explanation

At $\frac{3\pi}{2}$, the unit-circle point is at the bottom, giving a y-coordinate of $-1$ and an x-coordinate of zero. The pair $\sin x=0$ and $\cos x=-1$ instead occurs at $x=\pi$.

11. When the denominators are nonzero, how is tangent expressed in terms of sine and cosine?

A. $\tan x=\frac{1}{\cos x}$

B. $\tan x=\frac{1}{\sin x}$

C. $\tan x=\frac{\sin x}{\cos x}$

D. $\tan x=\frac{\cos x}{\sin x}$

Answer: (C)

Explanation

Tangent is defined as sine divided by cosine wherever cosine is nonzero. The reversed quotient is cotangent, while the two reciprocal expressions define cosecant and secant.

12. Which sign pattern is correct for the trigonometric functions in quadrant II?

A. Sine is negative and cosine is positive

B. Both sine and cosine are positive

C. Sine is positive and cosine is negative

D. Both sine and cosine are negative

Answer: (C)

Explanation

In quadrant II, the unit-circle y-coordinate is positive and the x-coordinate is negative, so sine is positive and cosine is negative. The pattern with both functions positive belongs to quadrant I.

13. If $\tan x$ is defined, which reciprocal identity can be used to relate tangent and secant?

A. $1+\tan^2x=\sec^2x$

B. $1+\cot^2x=\sec^2x$

C. $1+\tan^2x=\cosec^2x$

D. $1+\sec^2x=\tan^2x$

Answer: (A)

Explanation

The tangent-secant identity is $1+\tan^2x=\sec^2x$. The corresponding identity for cotangent uses cosecant, so replacing secant with cosecant changes the relationship.

14. Which function is undefined at odd multiples of $\frac{\pi}{2}$?

A. Cosecant

B. Tangent

C. Sine

D. Cosine

Answer: (B)

Explanation

Tangent equals sine divided by cosine, so it is undefined when cosine is zero at $x=(2n+1)\frac{\pi}{2}$. Sine is defined for every real number, while cosecant is excluded at integer multiples of $\pi$.

15. Which range belongs to the tangent function?

A. All real numbers

B. The interval $[-1,1]$

C. Values satisfying $y\leq-1$ or $y\geq1$

D. The interval $[0,1]$

Answer: (A)

Explanation

Tangent can take every real value, so its range is all real numbers. Sine and cosine are restricted to $[-1,1]$, while secant and cosecant have ranges outside the interval $(-1,1)$.

16. What is the fundamental period of $\tan x$?

A. $2\pi$

B. $\pi$

C. $4\pi$

D. $\frac{\pi}{2}$

Answer: (B)

Explanation

Tangent repeats after a horizontal shift of $\pi$, so its fundamental period is $\pi$. Sine, cosine, secant, and cosecant instead repeat after $2\pi$.

17. Which identity correctly expands $\sin(x-y)$?

A. $\cos x\cos y-\sin x\sin y$

B. $\sin x\cos y+\cos x\sin y$

C. $\cos x\cos y+\sin x\sin y$

D. $\sin x\cos y-\cos x\sin y$

Answer: (D)

Explanation

The sine difference identity subtracts the second cross-product term, giving $\sin(x-y)=\sin x\cos y-\cos x\sin y$. The plus sign belongs to the sine sum identity, not the difference identity.

18. Which expression is equivalent to $\cos(x+y)$?

A. $\sin x\cos y-\cos x\sin y$

B. $\sin x\cos y+\cos x\sin y$

C. $\cos x\cos y-\sin x\sin y$

D. $\cos x\cos y+\sin x\sin y$

Answer: (C)

Explanation

The cosine sum identity combines the product of cosines with the negative product of sines: $\cos(x+y)=\cos x\cos y-\sin x\sin y$. The plus sign between the squared-function products applies to $\cos(x-y)$ instead.

19. Which formula gives the sine double angle?

A. $\sin 2x=2\sin x\cos x$

B. $\sin 2x=1-2\sin^2x$

C. $\sin 2x=\sin^2x-\cos^2x$

D. $\sin 2x=4\sin^3x-3\sin x$

Answer: (A)

Explanation

The sine double-angle identity is the product formula $\sin 2x=2\sin x\cos x$. The squared expressions are equivalent forms for $\cos 2x$, while the cubic expression has the triple-angle structure.

20. For a value of $x$ where the expression is defined, which formula represents $\tan 2x$?

A. $\frac{\tan x}{1-\tan^2x}$

B. $\frac{2\tan x}{1+\tan^2x}$

C. $\frac{1-\tan^2x}{2\tan x}$

D. $\frac{2\tan x}{1-\tan^2x}$

Answer: (D)

Explanation

The tangent double-angle identity is $\tan 2x=\frac{2\tan x}{1-\tan^2x}$ when defined. The denominator with a plus sign is associated with a different tangent identity, not the double-angle formula.

21. Which expression is equivalent to $\cos 3x$?

A. $3\sin x-4\sin^3x$

B. $3\cos x-4\cos^3x$

C. $4\sin^3x-3\sin x$

D. $4\cos^3x-3\cos x$

Answer: (D)

Explanation

The cosine triple-angle identity is $\cos 3x=4\cos^3x-3\cos x$. The expression involving sine is the corresponding identity for $\sin 3x$, so it does not represent the cosine triple angle.