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Precalculus : Trigonometric Functions

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Example Questions

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Example Question #16 :Find The Value Of Any Of The Six Trigonometric Functions

Find the value of $\csc\theta$ .

$\theta=\frac{11\pi}{6}$

Possible Answers:

$\csc\theta=\frac{2\sqrt3}{3}$

$\csc\theta=-2$

$\csc\theta=\frac{-1}{2}$

$\csc\theta=2$

Correct answer:

$\csc\theta=-2$

解释,nation:

Since

$\csc\theta=\frac{1}{\sin\theta}$

we begin by finding the value of $\sin\theta$ .

$\sin\frac{11\pi}{6}=-\frac{1}{2}$ .

Then,

$\csc\frac{11\pi}{6}=\frac{1}{-1/2}=-2$

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Example Question #17 :Find The Value Of Any Of The Six Trigonometric Functions

Determine the value of: $\cot(30^{\circ})$

Possible Answers:

$\frac{\sqrt3}{3}$

$\sqrt3$

$\frac{\sqrt3}{4}$

$\frac{\sqrt3}{2}$

$3\sqrt3$

Correct answer:

$\sqrt3$

解释,nation:

确定的价值 cot(30) , simplify cotangent into sine and cosine.

$cot(30)=\frac{cos(30)}{sin(30)}= \frac{\frac{\sqrt3}{2}}{\frac{1}{2}}= {\frac{\sqrt3}{2}}\cdot{\frac{2}{1}}=\sqrt3$

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Example Question #18 :Find The Value Of Any Of The Six Trigonometric Functions

Find the value of $sec(\pi)+cos(-\pi)$ , if possible.

Possible Answers:

$\textup{The answer is not defined.}$

$\frac{\pi^2+1}{\pi}$

$\frac{1}{2}$

Correct answer:

解释,nation:

In order to solve $sec(\pi)+cos(-\pi)$ , split up the expression into 2 parts.

$sec(\pi)= \frac{1}{cos(\pi)}=\frac{1}{-1}=-1$

$cos(-\pi)= -1$

$sec(\pi)+cos(-\pi)= -1-1=-2$

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Example Question #19 :Find The Value Of Any Of The Six Trigonometric Functions

Compute $\sec(300^\circ)$ , if possible.

Possible Answers:

$\textup{The answer does not exist.}$

$-\frac{1}{2}$

$\frac{1}{2}$

Correct answer:

解释,nation:

Rewrite the expression in terms of cosine.

$sec(300)=\frac{1}{cos(300)}$

Evaluate the value of cos(300) 在第四象限。

$cos(300)= \frac{1}{2}$

Substitute it back to the simplified expression of sec(300) .

$sec(300)=\frac{1}{cos(300)}=\frac{1}{\frac{1}{2}}=2$

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Example Question #20 :Find The Value Of Any Of The Six Trigonometric Functions

Determine $csc(\frac{3\pi}{4})$

Possible Answers:

$\frac{2\sqrt 3}{3}$

$\frac{1}{\sqrt 2}$

$\sqrt 2$

Correct answer:

$\sqrt 2$

解释,nation:

Remember that:

$csc(\frac{3\pi}{4})=\frac{1}{sin(\frac{3\pi}{4})}=\frac{1}{\frac{1}{\sqrt 2}}=\sqrt 2$

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Example Question #21 :Find The Value Of Any Of The Six Trigonometric Functions

What is the value of $sin\left(\frac{\pi}{4}\right)$ ?

Possible Answers:

$\sqrt2$

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{\sqrt2}{2}$

Correct answer:

$\frac{\sqrt2}{2}$

解释,nation:

The sine of an angle corresponds to the y-component of the triangle in the unit circle. The angle $\frac{\pi}{4}$ is a special angle. In the unit circle, the hypotenuse is the radius of the unit circle, which is 1. Since the angle is $\frac{\pi}{4}$ , the triangle is an isosceles right triangle, or a 45-45-90.

Use the Pythagorean Theorem to solve for the leg. Both legs will be equal to each other.

a^2+b^2 = c^2

a^2+a^2=1^2

2a^2=1

$a=\sqrt\frac{1}{2}$

Rationalize the denominator.

$a=\frac{1}{\sqrt2}\cdot \frac{\sqrt2}{\sqrt2} = \frac{\sqrt2}{2}$

Therefore, $sin(\frac{\pi}{4})= \frac{\sqrt2}{2}$ .

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Example Question #1 :Prove Trigonometric Identities

Simplify: $1+\frac{cos^2(\theta)}{sin^2(\theta)}$

Possible Answers:

$sin^2(\theta)$

$sec^2(\theta)$

$csc^2(\theta)$

$\frac{1}{cot^2{\theta}}$

$\pi+tan^2(\theta)$

Correct answer:

$csc^2(\theta)$

解释,nation:

To simplify $1+\frac{cos^2(\theta)}{sin^2(\theta)}$ , find the common denominator and multiply the numerator accordingly.

$1+\frac{cos^2(\theta)}{sin^2(\theta)}=\frac{sin^2(\theta)}{sin^2(\theta)}+\frac{cos^2(\theta)}{sin^2(\theta)}=\frac{sin^2(\theta)+cos^2(\theta)}{sin^2(\theta)}$

The numerator is an identity.

$sin^2(\theta)+cos^2(\theta)=1$

Substitute the identity and simplify.

$\frac{sin^2(\theta)+cos^2(\theta)}{sin^2(\theta)}= \frac{1}{sin^2(\theta)}= csc^2(\theta)$

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Example Question #2 :Prove Trigonometric Identities

Evaluate in terms of sines and cosines:

$\frac{sec^2(\theta)}{cot(\theta)}$

Possible Answers:

$\frac{1}{cos(\theta)sin(\theta)}$

$\frac{sin(\theta)}{cos^3(\theta)}$

$\frac{sin(\theta)}{cos^2(\theta)}$

$\frac{sin(\theta)}{cos(\theta)}$

$\frac{cos(\theta)}{sin(\theta)}$

Correct answer:

$\frac{sin(\theta)}{cos^3(\theta)}$

解释,nation:

Convert $\frac{sec^2(\theta)}{cot(\theta)}$ into its sines and cosines.

$\frac{sec^2(\theta)}{cot(\theta)}= \left(\frac{1}{cos(\theta)}\times\frac{1}{cos(\theta)}\right)\div \frac{cos(\theta)}{sin(\theta)}$

$\left(\frac{1}{cos(\theta)}\times\frac{1}{cos(\theta)}\right)\div \frac{cos(\theta)}{sin(\theta)}=\frac{1}{cos^2(\theta)}\times\frac{sin(\theta)}{cos(\theta)}= \frac{sin(\theta)}{cos^3(\theta)}$

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Example Question #3 :Prove Trigonometric Identities

Simplify the following:

$sin^3(x)+sin(x)\cdot cos^2(x)$

Possible Answers:

The expression is already in simplified form

cos(x)

sin(x)

Correct answer:

sin(x)

解释,nation:

$sin^3(x)+sin(x)\cdot cos^2(x)$

First factor out sine x.

$=(sin^2(x)+cos^2(x))\cdot sin(x)$

Notice that a Pythagorean Identity is present.

The identity needed for this problem is:

(sin^2(x)+cos^2(x))=1

Using this identity the equation becomes,

$=(1)\cdot sin(x)$

=sin(x) .

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Example Question #4 :Prove Trigonometric Identities

Simplify the expression $\frac{sin(x)}{cos(x)}\times \frac{1}{cot(x)}$

Possible Answers:

tan^2(x)

sin^2(x)

cos^2(x)

csc^2(x)

Correct answer:

tan^2(x)

解释,nation:

To simplify, use the trigonometric identities $\frac{sin(x)}{cos(x)}=tan(x)$ and $\frac{1}{cot(x)}=tan(x)$ to rewrite both halves of the expression:

$tan(x)\times tan(x)$

Then combine using an exponent to simplify:

tan^2(x)

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Precalculus : Trigonometric Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #16 :Find The Value Of Any Of The Six Trigonometric Functions

Example Question #17 :Find The Value Of Any Of The Six Trigonometric Functions

Example Question #18 :Find The Value Of Any Of The Six Trigonometric Functions

Example Question #19 :Find The Value Of Any Of The Six Trigonometric Functions

Example Question #20 :Find The Value Of Any Of The Six Trigonometric Functions

Example Question #21 :Find The Value Of Any Of The Six Trigonometric Functions

Example Question #1 :Prove Trigonometric Identities

Example Question #2 :Prove Trigonometric Identities

Example Question #3 :Prove Trigonometric Identities

Example Question #4 :Prove Trigonometric Identities

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