Step-by-step explanation:
Since [tex]cos(\theta) = -\frac{3}{4}[/tex] we can get some information from this. First of all [tex]cos(\theta)[/tex] is defined as [tex]\frac{adjacent}{hypotenuse}[/tex]. So the adjacent side is 3 and the hypotenuse is 4. Using this we can find the opposite side to find [tex]sin(\theta)[/tex] to calculate csc and cot of theta. So using the Pythagorean Theorem we can solve for the missing side. Also I forget to mention we have to calculate the sign of the adjacent side and the hypotenuse. Since you're given that the angle is in quadrant 2, that means the x-value is going to be negative, and the y-value is going to be negative. And the x really represents the adjacent side and the y represents the opposite. So the adjacent side is what's negative. and the opposite is positive
Pythagorean Theorem:
[tex]a^2+b^2=c^2[/tex]
[tex](-3)^2 + b^2 = 4^2[/tex]
[tex]9 + b^2 = 16[/tex]
[tex]b^2 = 7[/tex]
[tex]b = \sqrt{7}[/tex]
So now we can calculate [tex]sin(\theta)[/tex].
[tex]sin(\theta) = \frac{\sqrt{7}}{4}[/tex]
Now to calculate the exact value of csc you simply take the inverse. This gives you [tex]csc(\theta)=\frac{4}{\sqrt{7}}[/tex]. Multiplying both sides by sqrt(7) to rational the denominator gives you [tex]\frac{4\sqrt7}{7}[/tex].
Now to calculate cot(theta) you find the inverse of tan. Tan is defined as [tex]tan(\theta) = \frac{sin(\theta)}{cos(\theta)}[/tex]. So all you do is take the inverse which is [tex]cot(\theta) = \frac{cos(\theta)}{sin(\theta)}[/tex].
Plug values in
[tex]cot(\theta) = \frac{-\frac{3}{4}}{\frac{4\sqrt{7}}{7}}[/tex]
Keep, change, flip
[tex]-\frac{3}{4} * \frac{7}{4\sqrt7}[/tex]
Multiply:
[tex]-\frac{21}{16\sqrt7}[/tex]
Multiply both sides by sqrt(7)
[tex]-\frac{21\sqrt7}{112}[/tex]
This is the value of cot(theta)
