An angle in standard position in the coordinate plane has a measure in radians of [tex]$\theta$[/tex], and its terminal side is in Quadrant IV. The value of [tex]$\cos \theta$[/tex] is [tex]$\frac{39}{89}$[/tex].

Part A

What is the value of [tex]$\sin \theta$[/tex]? Drag a number into the empty box to create your answer.

[tex]\[
\sin \theta = \boxed{}
\][/tex]

- [tex]$-\frac{89}{39}$[/tex]
- [tex]$-\frac{89}{80}$[/tex]
- [tex]$-\frac{80}{89}$[/tex]
- [tex]$-\frac{39}{89}$[/tex]
- [tex]$\frac{39}{89}$[/tex]
- [tex]$\frac{80}{89}$[/tex]
- [tex]$\frac{89}{80}$[/tex]
- [tex]$\frac{89}{39}$[/tex]

Part B

What is the value of [tex]$\tan \theta$[/tex]? Drag a number into the empty box to create your answer.

[tex]\[
\tan \theta = \boxed{}
\][/tex]

- [tex]$-\frac{89}{39}$[/tex]
- [tex]$-\frac{80}{39}$[/tex]
- [tex]$-\frac{39}{80}$[/tex]
- [tex]$-\frac{39}{89}$[/tex]
- [tex]$\frac{39}{89}$[/tex]
- [tex]$\frac{39}{80}$[/tex]
- [tex]$\frac{80}{39}$[/tex]
- [tex]$\frac{89}{39}$[/tex]



Answer :

To solve for the values of [tex]\(\sin \theta\)[/tex] and [tex]\(\tan \theta\)[/tex] given that [tex]\(\cos \theta = \frac{39}{89}\)[/tex] and the angle [tex]\(\theta\)[/tex] is in the fourth quadrant, we use trigonometric identities and the given trigonometric functions.

### Part A: Finding [tex]\(\sin \theta\)[/tex]
We know from the Pythagorean identity that:
[tex]\[ \cos^2 \theta + \sin^2 \theta = 1 \][/tex]

Given [tex]\(\cos \theta = \frac{39}{89}\)[/tex], we first find [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = \left(\frac{39}{89}\right)^2 = \frac{1521}{7921} \][/tex]

Next, we use the Pythagorean identity to solve for [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{1521}{7921} = \frac{7921 - 1521}{7921} = \frac{6400}{7921} \][/tex]

Then, taking the square root to find [tex]\(\sin \theta\)[/tex], we get:
[tex]\[ \sin \theta = \pm \sqrt{\frac{6400}{7921}} = \pm \frac{80}{89} \][/tex]

Since [tex]\(\theta\)[/tex] is in the fourth quadrant, [tex]\(\sin \theta\)[/tex] is negative:
[tex]\[ \sin \theta = -\frac{80}{89} \][/tex]

Thus, the value of [tex]\(\sin \theta\)[/tex] is:
[tex]\[ \boxed{-\frac{80}{89}} \][/tex]

### Part B: Finding [tex]\(\tan \theta\)[/tex]
Now, we use the definition of tangent in terms of sine and cosine:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]

We already have [tex]\(\sin \theta = -\frac{80}{89}\)[/tex] and [tex]\(\cos \theta = \frac{39}{89}\)[/tex]. Therefore:
[tex]\[ \tan \theta = \frac{-\frac{80}{89}}{\frac{39}{89}} = \frac{-80}{39} \][/tex]

Thus, the value of [tex]\(\tan \theta\)[/tex] is:
[tex]\[ \boxed{-\frac{80}{39}} \][/tex]