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4.) What is the exact value of sinθ when θ lies in Quadrant II and cosθ=−513
Fill in the blanks.
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10.) Suppose that a laser light, positioned 100 ft from the base of a flag pole, illuminates a flag that is 85 ft above the ground.

What is the angle of inclination (angle of elevation) of the light beam?

Express your answer in degrees rounded to the nearest hundredth.
Enter your answer in the box.

11.)

Suppose that a man standing at the edge of a cliff near the North Rim of the Grand Canyon is looking downward towards a campground inside the canyon. The elevation of the North Rim is 5389 ft and the elevation of the campground is 2405 ft. The man's range finder indicates that his line of sight distance to the campground is 3044 ft.

What is the angle of depression of the man's line of sight to the campground?

Express your answer in degrees rounded to the nearest hundredth.
Enter your answer in the box.

12.) What is the exact value of arcsin(0.5)?
Determine the value in degrees.

13.) Determine the exact value in degrees:
What is the exact value of arcsin−2√2

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Answer:

Part 4) [tex]sin(\theta)=\frac{12}{13}[/tex]

Part 10) The angle of elevation is [tex]40.36\°[/tex]

Part 11) The angle of depression is [tex]78.61\°[/tex]

Part 12) [tex]arcsin(0.5)=30\°[/tex]  or [tex]arcsin(0.5)=150\°[/tex]

Part 13) [tex]-45\°[/tex]  or [tex]225\°[/tex]

Step-by-step explanation:

Part 4) we have that

[tex]cos(\theta)=-\frac{5}{13}[/tex]

The angle theta lies in Quadrant II

so

The sine of angle theta is positive

Remember that

[tex]sin^{2}(\theta)+ cos^{2}(\theta)=1[/tex]

substitute the given value

[tex]sin^{2}(\theta)+(-\frac{5}{13})^{2}=1[/tex]

[tex]sin^{2}(\theta)+(\frac{25}{169})=1[/tex]

[tex]sin^{2}(\theta)=1-(\frac{25}{169})[/tex]  

[tex]sin^{2}(\theta)=(\frac{144}{169})[/tex]

[tex]sin(\theta)=\frac{12}{13}[/tex]

Part 10)

Let

[tex]\theta[/tex] ----> angle of elevation

we know that

[tex]tan(\theta)=\frac{85}{100}[/tex] ----> opposite side angle theta divided by adjacent side angle theta

[tex]\theta=arctan(\frac{85}{100})=40.36\°[/tex]

Part 11)

Let

[tex]\theta[/tex] ----> angle of depression

we know that

[tex]sin(\theta)=\frac{5,389-2,405}{3,044}[/tex] ----> opposite side angle theta divided by hypotenuse

[tex]sin(\theta)=\frac{2,984}{3,044}[/tex]

[tex]\theta=arcsin(\frac{2,984}{3,044})=78.61\°[/tex]

Part 12) What is the exact value of arcsin(0.5)?

Remember that

[tex]sin(30\°)=0.5[/tex]

therefore

[tex]arcsin(0.5)[/tex] -----> has two solutions

[tex]arcsin(0.5)=30\°[/tex] ----> I Quadrant

or

[tex]arcsin(0.5)=180\°-30\°=150\°[/tex] ----> II Quadrant

Part 13) What is the exact value of [tex]arcsin(-\frac{\sqrt{2}}{2})[/tex]

The sine is negative

so

The angle lies in Quadrant III or Quadrant IV

Remember that

[tex]sin(45\°)=\frac{\sqrt{2}}{2}[/tex]

therefore

[tex]arcsin(-\frac{\sqrt{2}}{2})[/tex] ----> has two solutions

[tex]arcsin(-\frac{\sqrt{2}}{2})=-45\°[/tex] ----> IV Quadrant

or

[tex]arcsin(-\frac{\sqrt{2}}{2})=180\°+45\°=225\°[/tex] ----> III Quadrant

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