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Precalculus : Evaluating Trig Functions

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Example Questions

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

3.2

2.3

2.9

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:

0.643

-0.156

0.577

3.125

1.732

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 #2 : Find The Decimal Value Of A Trigonometric Function

Find the decimal value of

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

Possible Answers:

0.866

0.3085

0.5

1.732

0.1542

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 #1 : Evaluating Trig Functions

Find the value of

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

Possible Answers:

-1.13

1.5

0.5

-0.5

1.13

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 #2 : 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:

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

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

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

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

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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Example Question #3 : Evaluating Trig Functions

Solve for all x on the interval $0^{\circ}\leq x \leq 90^{\circ}$

tan(x)=1

Possible Answers:

$45^{\circ}$

$30^{\circ}$

$0^{\circ}$

$90^{\circ}$

Correct answer:

$45^{\circ}$

Explanation:

Solve for all x on the interval $0^{\circ}\leq x \leq 90^{\circ}$

tan(x)=1

Remember Soh, Cah, Toa?

For this problem it helps to recall that $Tangent=\frac{opposite}{adjacent}$

Since our tangent is equal to 1 in this problem, we know that our opposite and adjacent sides must be the same (otherwise we wouldn't get "1" when we divided them)

Can you think of any angles in the first quadrant which yield equal x and y values?

If you guessed $45^{\circ}$ you guessed right! Remember that your $45^{\circ}$ angle in the unit circle will give you a 45/45/90 triangle, which will have equal height and base.

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Example Question #1 : Evaluating Trig Functions

Q3 new

The above triangle is a right triangle. Find the value of $\theta$ (in degrees).

Possible Answers:

$57^\circ$

$33^\circ$

$65^\circ$

$28^\circ$

$49^\circ$

Correct answer:

$57^\circ$

Explanation:

One can setup the relationship

$cos(\theta)=\frac{adjacent}{hypotenuse}$

$cos(\theta)=\frac{6}{11}$ .

After taking the arccosine,

$cos^{-1}(cos(\theta))=cos^{-1}\left(\frac{6}{11} \right )$

the arccosine cancels out the cosine leaving just the value of $\theta$ .

$\\ \theta=cos^{-1}\left(\frac{6}{11} \right ) \\ \theta= 56.944^\circ \\ \theta= 57^\circ$

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Example Question #1 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

What is the value of $\theta$ (in degrees)?

Possible Answers:

$57^\circ$

$61^\circ$

$42^\circ$

$70^\circ$

$26^\circ$

Correct answer:

$61^\circ$

Explanation:

One can setup the relationship

$\\tan(\theta) = \frac{opposite}{adjacent}\\ tan(\theta)=\frac{9}{5}$ .

After taking the arctangent,

$tan^{-1}(tan(\theta))=tan^{-1}\left(\frac{9}{5}\right)$

the arctangent cancels out the tangent and we are left with the value of $\theta$ .

$\\ \theta=tan^{-1}\left(\frac{9}{5} \right ) \\ \theta=60.945^\circ \\ \theta=61^\circ$

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Example Question #2 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

Solve for $\theta$ :

$\sin(3\theta) = -\frac{\sqrt{3}}{2}$

Possible Answers:

$\frac{ \pi }{9}$ or $\frac{2 \pi }{9}$

$\frac{5 \pi }{9}$ or $\frac{\pi }{9 }$

$\frac{5 \pi }{9}$ or $\frac{4 \pi }{9 }$

$5 \pi$ or $4 \pi$

$\frac{4 \pi}{3}$ or $\frac{5 \pi }{3 }$

Correct answer:

$\frac{5 \pi }{9}$ or $\frac{4 \pi }{9 }$

Explanation:

If the sine of an angle, in this case $3 \theta$ is $-\frac{ \sqrt 3 }{2}$ , the angle must be $\frac{4 \pi }{3}$ or $\frac{5 \pi }{3 }$ .

Then we need to solve for theta by dividing by 3:

$\frac{4 \pi }{3} \div 3 = \frac{4 \pi }{3} \cdot \frac{1}{3} = \frac{4 \pi }{9 }$

$\frac{5 \pi }{3} \div 3 = \frac{5 \pi }{3} \cdot \frac{1}{3} = \frac{5 \pi }{9 }$

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Precalculus : Evaluating Trig Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Evaluating Trig Functions

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

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

Example Question #2 : Evaluating Trig Functions

Example Question #3 : Evaluating Trig Functions

Example Question #1 : Evaluating Trig Functions

Example Question #1 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

Example Question #2 : Find The Degree Measure Of An Angle For Which The Value Of A Trigonometric Function Is Known

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