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Precalculus : Trigonometric Functions

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Example Questions

Example Question #1 : Proving Trig Identities

Simplify $\frac{cos(x)}{sin(x)}$ .

Possible Answers:

cot(x)

tan(x)

csc(x)

sec(x)

Correct answer:

cot(x)

Explanation:

This expression is a trigonometric identity: $\frac{cos(x)}{sin(x)}=cot(x)$

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Example Question #3 : Prove Trigonometric Identities

Simplify $\frac{2}{csc^2(x)}+\frac{2}{sec^2(x)}$

Possible Answers:

2csc(x)

csc(x)

Correct answer:

Explanation:

Factor out 2 from the expression:

$2(\frac{1}{csc^2(x)}+\frac{1}{sec^2(x)})$

Then use the trigonometric identities $\frac{1}{csc(x)}=sin(x)$ and $\frac{1}{sec(x)}=cos(x)$ to rewrite the fractions:

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

Finally, use the trigonometric identity sin^2(x)+cos^2(x)=1 to simplify:

2(1) = 2

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Example Question #257 : Pre Calculus

Simplify $\frac{2sin^2(x)}{tan(x)}+\frac{2cos^2(x)}{tan(x)}$

Possible Answers:

2cot(x)

4cot(x)

4tan(x)

2tan(x)

Correct answer:

2cot(x)

Explanation:

Factor out the common $\frac{2}{tan(x)}$ from the expression:

$(\frac{2}{tan(x)})(sin^2(x)+cos^2(x))$

Next, use the trigonometric identify sin^2(x)+cos^2(x)=1 to simplify:

$\frac{2}{tan(x)}(1)$

Then use the identify $\frac{1}{tan(x)}=cot(x)$ to simplify further:

2cot(x)

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Example Question #8 : Prove Trigonometric Identities

Simplify $\frac{2tan^2x+1}{tan^2x}+tan^2x$

Possible Answers:

cscx

sec^2x+csc^2x

2sec^2x

2csc^2x

Correct answer:

sec^2x+csc^2x

Explanation:

To simplify the expression, separate the fraction into two parts:

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

The tan^2x terms in the first fraction cancel leaving you with:

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

Then you can deal with the remaining fraction using the rule that $\frac{1}{tan^2x}=cot^2x$ . This leaves:

2+cot^2x+tan^2x

You can separate this into:

1+tan^2x+1+cot^2x

And each half of this expression is now a trigonometric identity: 1+tan^2x = sec^2x and 1+cot^2x=csc^2x . This gives you:

sec^2x+csc^2x

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Example Question #1 : Evaluating Trig Functions

Find the value of cot(30)+tan(30) to the nearest tenth if sin(30)=0.5 and cos(30)= 0.866 .

Possible Answers:

1.7

2.9

3.2

2.3

Correct answer:

2.3

Explanation:

Rewrite cot(30)+tan(30) in terms of sine and cosine.

$cot(30)+tan(30)=\frac{cos(30)}{sin(30)}+\frac{sin(30)}{cos(30)}$

Substitute the known values and evaluate.

$\frac{0.866}{0.5}+\frac{0.5}{0.866}=1.732+0.577367 =2.309367$

The answer to the nearest tenth is 2.3 .

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Example Question #1 : Find The Decimal Value Of A Trigonometric Function

Determine the value of $cot(\frac{\pi}{3})$ in decimal form.

Possible Answers:

1.732

0.643

3.125

0.577

-0.156

Correct answer:

0.577

Explanation:

Ensure the calculator is in radian mode since the expression shows the angle in terms of $\pi$ . Also convert cotangent to tangent.

$cot(\frac{\pi}{3})=\frac{1}{tan(\frac{\pi}{3})}= \frac{1}{\sqrt3}=0.577$

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Example Question #1 : Evaluating Trig Functions

Find the decimal value of

$2\cos\left(\frac{\pi}{6}\right)$

Possible Answers:

0.5

0.3085

0.1542

0.866

1.732

Correct answer:

1.732

Explanation:

To determine the decimal value of the following trig function, $2cos(\frac{\pi}{6})$ , make sure that the calculator is in radian mode.

Compute the expression.

$2cos(\frac{\pi}{6})=2(0.866)=1.732$

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Example Question #2 : Find The Decimal Value Of A Trigonometric Function

Determine the correct value of

$sin\left(\frac{1}{2}^{\circ}\right)$ .

Possible Answers:

2.62

1.05

0.48

0.009

1.57

Correct answer:

0.009

Explanation:

The question $sin(\frac{1}{2})$ asks for the y-coordinate on the unit circle when the degree angle is $\frac{1}{2}$ .

Be careful not to confuse finding the value of the angle when the y-value of the coordinate of the unit circle is $\frac{1}{2}$ .

Ensure that the calculator is in degree mode.

$sin(\frac{1}{2})=0.00872=0.009$

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Example Question #2 : Find The Decimal Value Of A Trigonometric Function

Find the value of

$\sin(270^\circ)+\cos(60^\circ)$ .

Possible Answers:

1.13

0.5

-1.13

-0.5

1.5

Correct answer:

-0.5

Explanation:

Before beginning this problem on a calculator, though this is not necessary since these are special angles, ensure that the mode of the calculator is in degrees.

Input the values of the expression and solve.

sin(270)+cos(60)= -1+0.5 = -0.5

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Example Question #1 : Evaluating Trig Functions

Solve for all x on the interval $0^{\circ} \leq x \leq 360^{\circ}$

$sin(x)=\frac{\sqrt{2}}{2}$

Possible Answers:

$45^{\circ}$ , $90^{\circ}$

$90^{\circ}$ , $135^{\circ}$

$135^{\circ}$ , $225^{\circ}$

$45^{\circ}$ , $135^{\circ}$

Correct answer:

$45^{\circ}$ , $135^{\circ}$

Explanation:

Solve for all x on the interval $0^{\circ} \leq x \leq 360^{\circ}$

$sin(x)=\frac{\sqrt{2}}{2}$

We can begin by recalling which two quadrants have a positive sine. Because sine corresponds to the y-value, we know that sine is positive in quadrants I and II.

Next, recall where we get $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2}$ always corresponds to our $45^{\circ}$ -increment angles. In this case, the angles we are looking for are $45^{\circ}$ and $135^{\circ}$ , because those are the two $45^{\circ}$ -increment angles in the first two quadrants.

Now, you might be saying, "what about $90^{\circ}$ ? That is an increment of 45."

While that is true, $sin(90^{\circ})=1$ , not $\frac{\sqrt{2}}{2}$

So our answer is:

$45^{\circ}$ , $135^{\circ}$

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Precalculus : Trigonometric Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Proving Trig Identities

Example Question #3 : Prove Trigonometric Identities

Example Question #257 : Pre Calculus

Example Question #8 : Prove Trigonometric Identities

Example Question #1 : Evaluating Trig Functions

Example Question #1 : Find The Decimal Value Of A Trigonometric Function

Example Question #1 : Evaluating Trig Functions

Example Question #2 : Find The Decimal Value Of A Trigonometric Function

Example Question #2 : Find The Decimal Value Of A Trigonometric Function

Example Question #1 : Evaluating Trig Functions

All Precalculus Resources

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