Question 1

If $\sin x = \frac{3}{5}$ and $x$ is in the second quadrant, what is $\cos x$?

Accepted Answer

Correct Option: B. Solution: In the second quadrant, cosine is negative. Using $\sin^2 x + \cos^2 x = 1$, we find $\cos x = -\frac{4}{5}$.

Question 2

What is the correct expression for $\sin(2x)$ using double angle formulas?

Accepted Answer

Correct Option: A. Solution: The double angle formula for $\sin(2x)$ is $2 \sin x \cos x$.

Question 3

Consider the function $y = \sin(x)$. If the function is shifted to the right by $\frac{\pi}{2}$ units, what is the new equation of the function?

Accepted Answer

Correct Option: B. Solution: A right shift by $\frac{\pi}{2}$ units in the graph of $y = \sin(x)$ results in $y = \sin(x - \frac{\pi}{2})$, which is equivalent to $y = \cos(x)$.

Question 4

Given the identity $$\sin(x + y) = \sin x \cos y + \cos x \sin y$$, which of the following expressions correctly represents $$\sin(x - y)$$?

Accepted Answer

Correct Option: A. Solution: The identity for $$\sin(x - y)$$ is derived by replacing $$y$$ with $$-y$$ in the identity for $$\sin(x + y)$$, resulting in $$\sin(x - y) = \sin x \cos y - \cos x \sin y$$.

Question 5

If $$\cos x = -\frac{3}{5}$$ and $$x$$ lies in the third quadrant, what is the value of $$\sin(x + 45^	ext{o})$$?

Accepted Answer

Correct Option: A. Solution: In the third quadrant, $$\sin x$$ is negative. Since $$\cos^2 x + \sin^2 x = 1$$, we have $$\sin x = -\frac{4}{5}$$. Using the identity $$\sin(x + y) = \sin x \cos y + \cos x \sin y$$, and substituting $$y = 45^	ext{o}$$, $$\sin(x + 45^	ext{o}) = -\frac{4}{5} \cdot \frac{1}{\sqrt{2}} + (-\frac{3}{5}) \cdot \frac{1}{\sqrt{2}} = -\frac{4}{5}$$.

Question 6

If $$	an x = \frac{3}{4}$$ and $$x$$ lies in the first quadrant, what is the value of $$\sec x$$?

Accepted Answer

Correct Option: C. Solution: Using the identity $$1 + 	an^2 x = \sec^2 x$$, we find $$\sec x = \sqrt{1 + \left(\frac{3}{4}ight)^2} = \frac{5}{4}$$.

Question 7

If $\sin x = \frac{3}{5}$ and $x$ lies in the second quadrant, what is $\cos x$?

Accepted Answer

Correct Option: B. Solution: In the second quadrant, cosine is negative. Using $\sin^2 x + \cos^2 x = 1$, we find $\cos x = -\frac{4}{5}$.

Question 8

What is the sign of $	an x$ in the third quadrant?

Accepted Answer

Correct Option: A. Solution: In the third quadrant, both sine and cosine are negative, making the tangent positive since $	an x = \frac{\sin x}{\cos x}$.

Question 9

What is the domain of the function $y = 	an x$?

Accepted Answer

Correct Option: B. Solution: The function $y = 	an x$ is undefined at $x = (2n + 1)\frac{\pi}{2}$, where $n$ is an integer, due to vertical asymptotes.

Question 10

Which of the following is the correct expression for the cosine of the sum of two angles?

Accepted Answer

Correct Option: A. Solution: The identity for the cosine of the sum of two angles is $\cos(x + y) = \cos x \cos y - \sin x \sin y$.

Question 11

Which of the following is the correct expression for $\cos(x + y)$?

Accepted Answer

Correct Option: B. Solution: The expression for $\cos(x + y)$ is $\cos x \cos y - \sin x \sin y$ according to the angle addition formula for cosine.

Question 12

What is the amplitude of the function $y = 3\cos(x) + 2$?

Accepted Answer

Correct Option: A. Solution: The amplitude of a cosine function $y = a\cos(x) + b$ is the absolute value of $a$. Here, the amplitude is $|3| = 3$.

Question 13

Which of the following represents the correct transformation for the identity $$\cos(2x) = 2\cos^2 x - 1$$?

Accepted Answer

Correct Option: A. Solution: Using the Pythagorean identity $$\sin^2 x + \cos^2 x = 1$$, we can transform $$\cos(2x) = 2\cos^2 x - 1$$ into $$\cos(2x) = 1 - 2\sin^2 x$$.

Question 14

If $	an x = \frac{3}{4}$ and $x$ lies in the third quadrant, what is the value of $\sin x$?

Accepted Answer

Correct Option: B. Solution: In the third quadrant, both sine and tangent are negative. Using $1 + 	an^2 x = \sec^2 x$, we find $\sec x = -\frac{5}{4}$. Then $\cos x = -\frac{4}{5}$ and $\sin x = -\frac{3}{5}$.

Question 15

A circle has a radius of 10 cm. What is the length of an arc that subtends an angle of $\frac{\pi}{3}$ radians at the center?

Accepted Answer

Correct Option: C. Solution: The length of an arc $l$ is given by $l = r 	heta$, where $r$ is the radius and $	heta$ is the angle in radians. Here, $l = 10 	imes \frac{\pi}{3} = \frac{10\pi}{3}$ cm, which is approximately 30 cm.

Question 16

What is the radian measure of an angle that subtends an arc of length equal to the radius of the circle?

Accepted Answer

Correct Option: A. Solution: By definition, an angle subtended by an arc of length equal to the radius in a unit circle is $1$ radian.

Question 17

What is the period of the sine function $\sin(x)$?

Accepted Answer

Correct Option: B. Solution: The period of the sine function $\sin(x)$ is $2\pi$, meaning it repeats its values every $2\pi$ units.

Question 18

Using the identity for $	an(x+y)$, what is the expression for $	an(75^\circ)$ if $	an 45^\circ = 1$ and $	an 30^\circ = \frac{1}{\sqrt{3}}$?

Accepted Answer

Correct Option: D. Solution: Using $	an(x+y) = \frac{	an x + 	an y}{1 - 	an x 	an y}$, for $x = 45^\circ$ and $y = 30^\circ$, $	an 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 	imes \frac{1}{\sqrt{3}}} = \frac{2 + \sqrt{3}}{1 + \sqrt{3}}$.

Question 19

Which of the following identities correctly represents the product-to-sum formula for $\cos x + \cos y$?

Accepted Answer

Correct Option: A. Solution: The product-to-sum formula for $\cos x + \cos y$ is $\cos x + \cos y = 2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$.

Question 20

What is the domain of the function $y = 	an x$?

Accepted Answer

Correct Option: A. Solution: The tangent function is undefined at $x = (2n + 1) \frac{\pi}{2}$ due to vertical asymptotes.

Question 21

What is the general solution for the equation $	an(x) = \sqrt{3}$?

Accepted Answer

Correct Option: A. Solution: The principal value of $	an^{-1}(\sqrt{3})$ is $\frac{\pi}{3}$. The general solution for $	an(x) = 	an(\frac{\pi}{3})$ is $x = n\pi + \frac{\pi}{3}$, where $n \in \mathbb{Z}$.

Question 22

Using the sum-to-product identities, what is the expression for $\sin x + \sin y$?

Accepted Answer

Correct Option: A. Solution: The sum-to-product identity for $\sin x + \sin y$ is $2 \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$.

Question 23

Which of the following is the period of the function $y = \sec(x)$?

Accepted Answer

Correct Option: A. Solution: The period of the secant function, $y = \sec(x)$, is $\pi$, as it repeats its values every $\pi$ units.

Question 24

If $\cos x = -\frac{3}{5}$ and $x$ lies in the third quadrant, what is the value of $\sin x$?

Accepted Answer

Correct Option: B. Solution: In the third quadrant, sine is negative. Using $\sin^2 x + \cos^2 x = 1$, we find $\sin x = -\frac{4}{5}$.

Question 25

Determine the angle in degrees for an angle of $2.5$ radians.

Accepted Answer

Correct Option: A. Solution: To convert radians to degrees, use the formula $	ext{degrees} = \frac{180}{\pi} 	imes 	ext{radians}$. Therefore, $2.5$ radians = $\frac{180}{\pi} 	imes 2.5 \approx 143°$.

Question 26

Convert the angle $-1710^\circ$ to a positive angle in the standard position.

Accepted Answer

Correct Option: A. Solution: The angle $-1710^\circ$ can be converted to a positive angle by adding $5 	imes 360^\circ$ to it, resulting in $-1710^\circ + 1800^\circ = 90^\circ$. Therefore, the angle in standard position is $90^\circ$.

Question 27

Given $$	an x = \frac{5}{12}$$ and $$x$$ lies in the second quadrant, find $$\sin(2x)$$.

Accepted Answer

Correct Option: D. Solution: In the second quadrant, $$\sin x$$ is positive and $$\cos x$$ is negative. Using $$	an x = \frac{5}{12}$$, we find $$\sin x = \frac{5}{13}$$ and $$\cos x = -\frac{12}{13}$$. Then, $$\sin(2x) = 2 \sin x \cos x = 2 \cdot \frac{5}{13} \cdot -\frac{12}{13} = -\frac{120}{169}$$.

Question 28

In which quadrant is the sine of an angle positive and the cosine negative?

Accepted Answer

Correct Option: B. Solution: In the second quadrant, sine is positive and cosine is negative as the angle lies between $\frac{\pi}{2}$ and $\pi$.

Question 29

Using the sum-to-product identities, what is the expression for $\sin x - \sin y$?

Accepted Answer

Correct Option: A. Solution: The sum-to-product identity for $\sin x - \sin y$ is $2 \cos \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)$.

Question 30

Evaluate the expression $\cos(75^\circ) + \cos(15^\circ)$ using product-to-sum identities.

Accepted Answer

Correct Option: A. Solution: Using the product-to-sum identity, $\cos(75^\circ) + \cos(15^\circ) = 2 \cos(45^\circ) \cos(30^\circ)$.

Question 31

If $\cos(x) = -\frac{3}{5}$ and $x$ is in the third quadrant, find $\sin(x)$.

Accepted Answer

Correct Option: B. Solution: In the third quadrant, both sine and cosine are negative. Using $\sin^2(x) + \cos^2(x) = 1$, we find $\sin(x) = -\sqrt{1 - \cos^2(x)} = -\frac{4}{5}$.

Question 32

Which of the following is the correct expression for $	an(x + y)$?

Accepted Answer

Correct Option: A. Solution: The formula for the tangent of the sum of two angles is $	an(x + y) = \frac{	an x + 	an y}{1 - 	an x 	an y}$.

Question 33

If the cosine of an angle is negative and the angle lies in the second quadrant, which of the following is true about the sine of the angle?

Accepted Answer

Correct Option: A. Solution: In the second quadrant, the sine function is positive while the cosine function is negative.

Question 34

Evaluate $\cos(3x)$ given that $\cos(x) = \frac{1}{2}$ and $x$ is in the first quadrant.

Accepted Answer

Correct Option: B. Solution: Using the triple angle formula for cosine, $\cos(3x) = 4\cos^3(x) - 3\cos(x)$. Substituting $\cos(x) = \frac{1}{2}$, we have $\cos(3x) = 4\left(\frac{1}{2}\right)^3 - 3\left(\frac{1}{2}\right) = -\frac{1}{2}$.

Question 35

If an angle $x$ is in the third quadrant, which of the following is true about $\sin x$ and $\cos x$?

Accepted Answer

Correct Option: C. Solution: In the third quadrant, both sine and cosine values are negative.

Question 36

Which of the following represents the range of the function $y = \sec x$?

Accepted Answer

Correct Option: A. Solution: The secant function has a range of $(-\infty, -1] \cup [1, \infty)$ as it is the reciprocal of the cosine function.

Question 37

If an angle measures $\frac{7\pi}{4}$ radians, what is its equivalent in degrees?

Accepted Answer

Correct Option: B. Solution: To convert radians to degrees, use the formula $	ext{degrees} = \frac{180}{\pi} 	imes 	ext{radians}$. Therefore, $\frac{7\pi}{4}$ radians = $\frac{180}{\pi} 	imes \frac{7\pi}{4} = 315°$.

Question 38

What is the value of $\sin(\pi - x)$?

Accepted Answer

Correct Option: A. Solution: The identity $\sin(\pi - x) = \sin x$ is derived from the properties of sine function in the unit circle.

Question 39

What is the general solution for the equation $\sin x = 0$?

Accepted Answer

Correct Option: A. Solution: The sine function is zero at integer multiples of $\pi$, hence the general solution is $x = n\pi$, where $n$ is an integer.

Question 40

Using the sum and difference identities, what is the value of $$	an(75^	ext{o})$$?

Accepted Answer

Correct Option: A. Solution: Using the identity $$	an(x + y) = \frac{	an x + 	an y}{1 - 	an x 	an y}$$, let $$x = 45^	ext{o}$$ and $$y = 30^	ext{o}$$. Then, $$	an(75^	ext{o}) = \frac{	an 45^	ext{o} + 	an 30^	ext{o}}{1 - 	an 45^	ext{o} 	an 30^	ext{o}} = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 	imes \frac{1}{\sqrt{3}}} = 2 + \sqrt{3}$$.

Question 41

Convert an angle of $60^\circ$ into radians.

Accepted Answer

Correct Option: A. Solution: Using the conversion $\pi$ radians = $180^\circ$, $60^\circ$ is equivalent to $\frac{60}{180}\pi = \frac{\pi}{3}$ radians.

Question 42

If $	an x = \frac{3}{4}$ and $x$ is in the first quadrant, what is $	an(2x)$?

Accepted Answer

Correct Option: A. Solution: Using the identity $	an(2x) = \frac{2 	an x}{1 - 	an^2 x}$, we have $	an(2x) = \frac{2 \cdot \frac{3}{4}}{1 - \left(\frac{3}{4}ight)^2} = \frac{\frac{6}{4}}{1 - \frac{9}{16}} = \frac{\frac{6}{4}}{\frac{7}{16}} = \frac{24}{7}$.

Question 43

Determine the value of $\sin(2x)$ if $\cos(x) = \frac{5}{13}$ and $x$ is in the fourth quadrant.

Accepted Answer

Correct Option: B. Solution: In the fourth quadrant, sine is negative. Using $\sin^2(x) + \cos^2(x) = 1$, we find $\sin(x) = -\frac{12}{13}$. Then $\sin(2x) = 2\sin(x)\cos(x) = 2(-\frac{12}{13})(\frac{5}{13}) = -\frac{120}{169}$.

Question 44

What is the value of $\sin(45^\circ - 30^\circ)$ using the sum and difference identities?

Accepted Answer

Correct Option: D. Solution: Using the identity $\sin(x-y) = \sin x \cos y - \cos x \sin y$, we have $\sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$.

Question 45

If $$\sin x = \frac{5}{13}$$ and $$x$$ is in the second quadrant, what is the value of $$\cos x$$?

Accepted Answer

Correct Option: B. Solution: In the second quadrant, cosine is negative. Using $$\sin^2 x + \cos^2 x = 1$$, we find $$\cos x = -\sqrt{1 - \left(\frac{5}{13}\right)^2} = -\frac{12}{13}$$.

Question 46

Which of the following expressions correctly represents $$\sin(3x)$$ using trigonometric identities?

Accepted Answer

Correct Option: A. Solution: The identity for $$\sin(3x)$$ is derived as $$\sin(3x) = 3\sin x - 4\sin^3 x$$ using the expansion of $$\sin(2x + x)$$.

Question 47

If $$	an x = \frac{3}{4}$$ and $$x$$ lies in the first quadrant, what is the value of $$	an(2x)$$?

Accepted Answer

Correct Option: A. Solution: Using the identity $$	an(2x) = \frac{2	an x}{1 - 	an^2 x}$$, substitute $$	an x = \frac{3}{4}$$ to get $$	an(2x) = \frac{2 	imes \frac{3}{4}}{1 - (\frac{3}{4})^2} = \frac{\frac{6}{4}}{1 - \frac{9}{16}} = \frac{24}{7}$$.

Question 48

If $\cos(x) = -3/5$ and $x$ lies in the third quadrant, what are the values of the other trigonometric functions?

Accepted Answer

Correct Option: A. Solution: In the third quadrant, both sine and cosine are negative. Given $\cos(x) = -3/5$, using $\sin^2(x) + \cos^2(x) = 1$, we find $\sin(x) = -4/5$. Thus, $	an(x) = \sin(x)/\cos(x) = 4/3$. The other functions follow as $\csc(x) = -5/4$, $\sec(x) = -5/3$, $\cot(x) = 3/4$.

Question 49

If $y = 	an(x)$, which of the following describes the behavior of the function as $x$ approaches $\frac{\pi}{2}$ from the left?

Accepted Answer

Correct Option: C. Solution: As $x$ approaches $\frac{\pi}{2}$ from the left, $	an(x)$ approaches $+\infty$ because the tangent function has a vertical asymptote at $x = \frac{\pi}{2}$.

Question 50

If $	an(x) = \frac{3}{4}$, what is the value of $	an(2x)$?

Accepted Answer

Correct Option: A. Solution: Using the formula $	an(2x) = \frac{2 	an x}{1 - 	an^2 x}$, we substitute $	an x = \frac{3}{4}$ to get $	an(2x) = \frac{2 	imes \frac{3}{4}}{1 - (\frac{3}{4})^2} = \frac{24}{7}$.

Question 51

Given that $\sin(x) = \frac{3}{5}$ and $\cos(y) = -\frac{5}{13}$, where $x$ and $y$ are angles in the second quadrant, find $\sin(x + y)$.

Accepted Answer

Correct Option: A. Solution: Using the identity $\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)$, we find $\cos(x) = -\frac{4}{5}$ and $\sin(y) = \frac{12}{13}$. Substituting these values, $\sin(x + y) = \frac{3}{5} 	imes -\frac{5}{13} + -\frac{4}{5} 	imes \frac{12}{13} = \frac{56}{65}$.

Question 52

In which quadrant is the sine function negative and the cosine function positive?

Accepted Answer

Correct Option: D. Solution: In the fourth quadrant, the sine function is negative and the cosine function is positive.

Question 53

What is the value of $\sin(31\pi/3)$ using the periodicity of trigonometric functions?

Accepted Answer

Correct Option: A. Solution: The value of $\sin(31\pi/3)$ can be simplified using the periodicity of the sine function. Since $\sin(x)$ repeats every $2\pi$, we have $\sin(31\pi/3) = \sin(31\pi/3 - 10\pi) = \sin(\pi/3) = \sqrt{3}/2$.

Question 54

If $\sin(x) = \frac{5}{13}$ and $x$ is in the second quadrant, find $\cos(2x)$.

Accepted Answer

Correct Option: B. Solution: Using the double angle formula for cosine, $\cos(2x) = 1 - 2\sin^2(x)$. Substituting $\sin(x) = \frac{5}{13}$, we find $\cos(2x) = 1 - 2\left(\frac{5}{13}\right)^2 = -\frac{119}{169}$.

Question 55

Which of the following is the correct triple angle formula for $\sin(3x)$?

Accepted Answer

Correct Option: A. Solution: The triple angle formula for $\sin(3x)$ is $3 \sin x - 4 \sin^3 x$.

Question 56

Which of the following trigonometric functions has a period of $2\pi$?

Accepted Answer

Correct Option: A. Solution: The sine function $\sin x$ has a period of $2\pi$, meaning it repeats its values every $2\pi$ radians.

Question 57

What is the product-to-sum identity for $2 \cos x \cos y$?

Accepted Answer

Correct Option: B. Solution: The product-to-sum identity for $2 \cos x \cos y$ is $\cos(x+y) + \cos(x-y)$.

Question 58

Which of the following is a fundamental identity in trigonometry?

Accepted Answer

Correct Option: D. Solution: All listed identities are fundamental trigonometric identities.

Question 59

If $	an x = -\frac{3}{4}$ and $x$ lies in the second quadrant, what is the value of $\sin x$?

Accepted Answer

Correct Option: A. Solution: In the second quadrant, sine is positive. Using the identity $1 + 	an^2 x = \sec^2 x$, we find $\sec x = \frac{5}{4}$. Since $\sec x = \frac{1}{\cos x}$, $\cos x = \frac{4}{5}$. Using $\sin^2 x + \cos^2 x = 1$, we find $\sin x = \frac{3}{5}$.

Question 60

Using the identity $$\cos(x + y) = \cos x \cos y - \sin x \sin y$$, which of the following is the correct expression for $$\cos(2x)$$?

Accepted Answer

Correct Option: D. Solution: The identity for $$\cos(2x)$$ can be derived from $$\cos(x + y)$$ by setting $$y = x$$. Thus, $$\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$$.

Question 61

If $$\cot x = 2$$ and $$x$$ lies in the fourth quadrant, what is the value of $$\cos(x - 30^	ext{o})$$?

Accepted Answer

Correct Option: C. Solution: In the fourth quadrant, $$\sin x$$ is negative and $$\cos x$$ is positive. Since $$\cot x = 2$$, we have $$	an x = \frac{1}{2}$$. Using the identity $$\cos(x - y) = \cos x \cos y + \sin x \sin y$$, and substituting $$y = 30^	ext{o}$$, we get $$\cos(x - 30^	ext{o}) = \cos x \cdot \frac{\sqrt{3}}{2} + \sin x \cdot \frac{1}{2} = -\frac{\sqrt{3}}{2}$$.

Question 62

Convert an angle of 150° to radians.

Accepted Answer

Correct Option: A. Solution: To convert degrees to radians, use the formula $	ext{radians} = \frac{\pi}{180} 	imes 	ext{degrees}$. Therefore, $150° = \frac{\pi}{180} 	imes 150 = \frac{5\pi}{6}$ radians.

Question 63

If $	an(x) = \frac{3}{4}$ and $x$ is in the first quadrant, find $	an(2x)$.

Accepted Answer

Correct Option: A. Solution: Using the double angle formula for tangent, $	an(2x) = \frac{2	an(x)}{1 - 	an^2(x)}$. Substituting $	an(x) = \frac{3}{4}$, we get $	an(2x) = \frac{2 	imes \frac{3}{4}}{1 - \left(\frac{3}{4}ight)^2} = \frac{24}{7}$.

Question 64

Which of the following is the range of the function $y = \sec x$?

Accepted Answer

Correct Option: C. Solution: The range of $y = \sec x$ is $(-\infty, -1] \cup [1, \infty)$, as $\sec x$ is undefined between $-1$ and $1$.

Trigonometric Functions

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Trigonometric Functions and Identities

Angle Measurement and Conversion

Graphical Representation of Trigonometric Functions

Domain and Range of Trigonometric Functions

Trigonometric Equations and Solutions

Trigonometric Functions of Multiple Angles

Signs and Quadrants of Trigonometric Functions

Periodicity and Reduction of Angles

Sum and Difference Identities

Product-to-Sum and Sum-to-Product Identities

Trigonometric Identity Proofs and Simplification

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Trigonometric Functions and Identities

Angle Measurement and Conversion

Graphical Representation of Trigonometric Functions

Domain and Range of Trigonometric Functions

Trigonometric Equations and Solutions

Trigonometric Functions of Multiple Angles

Signs and Quadrants of Trigonometric Functions

Periodicity and Reduction of Angles

Sum and Difference Identities

Product-to-Sum and Sum-to-Product Identities

Trigonometric Identity Proofs and Simplification

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Multiple Choice Questions

Q1.If sin⁡x=35\sin x = \frac{3}{5}sinx=53​ and xxx is in the second quadrant, what is cos⁡x\cos xcosx?

Correct Answer: B

Q2.What is the correct expression for sin⁡(2x)\sin(2x)sin(2x) using double angle formulas?

Correct Answer: A

Q3.Consider the function y=sin⁡(x)y = \sin(x)y=sin(x). If the function is shifted to the right by π2\frac{\pi}{2}2π​ units, what is the new equation of the function?

Correct Answer: B

Q4.Given the identity sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin(x + y) = \sin x \cos y + \cos x \sin ysin(x+y)=sinxcosy+cosxsiny, which of the following expressions correctly represents sin⁡(x−y)\sin(x - y)sin(x−y)?

Correct Answer: A

Q5.If cos⁡x=−35\cos x = -\frac{3}{5}cosx=−53​ and xxx lies in the third quadrant, what is the value of sin⁡(x+45exto)\sin(x + 45^ ext{o})sin(x+45exto)?

Correct Answer: A

Q6.If tan⁡x=34\tan x = \frac{3}{4}tanx=43​ and xxx lies in the first quadrant, what is the value of sec⁡x\sec xsecx?

Correct Answer: C

Q7.If sin⁡x=35\sin x = \frac{3}{5}sinx=53​ and xxx lies in the second quadrant, what is cos⁡x\cos xcosx?

Correct Answer: B

Q8.What is the sign of tan⁡x\tan xtanx in the third quadrant?

Correct Answer: A

Q9.What is the domain of the function y=tan⁡xy = \tan xy=tanx?

Correct Answer: B

Q10.Which of the following is the correct expression for the cosine of the sum of two angles?

Correct Answer: A

Q11.Which of the following is the correct expression for cos⁡(x+y)\cos(x + y)cos(x+y)?

Correct Answer: B

Q12.What is the amplitude of the function y=3cos⁡(x)+2y = 3\cos(x) + 2y=3cos(x)+2?

Correct Answer: A

Q13.Which of the following represents the correct transformation for the identity cos⁡(2x)=2cos⁡2x−1\cos(2x) = 2\cos^2 x - 1cos(2x)=2cos2x−1?

Correct Answer: A

Q14.If tan⁡x=34\tan x = \frac{3}{4}tanx=43​ and xxx lies in the third quadrant, what is the value of sin⁡x\sin xsinx?

Correct Answer: B

Q15.A circle has a radius of 10 cm. What is the length of an arc that subtends an angle of π3\frac{\pi}{3}3π​ radians at the center?

Correct Answer: C

Q16.What is the radian measure of an angle that subtends an arc of length equal to the radius of the circle?

Correct Answer: A

Q17.What is the period of the sine function sin⁡(x)\sin(x)sin(x)?

Correct Answer: B

Q18.Using the identity for tan⁡(x+y)\tan(x+y)tan(x+y), what is the expression for tan⁡(75∘)\tan(75^\circ)tan(75∘) if tan⁡45∘=1\tan 45^\circ = 1tan45∘=1 and tan⁡30∘=13\tan 30^\circ = \frac{1}{\sqrt{3}}tan30∘=3​1​?

Correct Answer: D

Q19.Which of the following identities correctly represents the product-to-sum formula for cos⁡x+cos⁡y\cos x + \cos ycosx+cosy?

Correct Answer: A

Q20.What is the domain of the function y=tan⁡xy = \tan xy=tanx?

Correct Answer: A

Q21.What is the general solution for the equation tan⁡(x)=3\tan(x) = \sqrt{3}tan(x)=3​?

Correct Answer: A

Q22.Using the sum-to-product identities, what is the expression for sin⁡x+sin⁡y\sin x + \sin ysinx+siny?