Question

Let theta be an angle in quadrant 2 such that cos theta =-3/4. Find the exact values of csc theta and cot theta.

Answer 1

Step-by-step explanation:

Since $cos(\theta) = -\frac{3}{4}$ we can get some information from this. First of all $cos(\theta)$ is defined as $\frac{adjacent}{hypotenuse}$. So the adjacent side is 3 and the hypotenuse is 4. Using this we can find the opposite side to find $sin(\theta)$ 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:

$a^2+b^2=c^2$

$(-3)^2 + b^2 = 4^2$

$9 + b^2 = 16$

$b^2 = 7$

$b = \sqrt{7}$

So now we can calculate $sin(\theta)$.

$sin(\theta) = \frac{\sqrt{7}}{4}$

Now to calculate the exact value of csc you simply take the inverse. This gives you $csc(\theta)=\frac{4}{\sqrt{7}}$. Multiplying both sides by sqrt(7) to rational the denominator gives you $\frac{4\sqrt7}{7}$.

Now to calculate cot(theta) you  find the inverse of tan. Tan is defined as $tan(\theta) = \frac{sin(\theta)}{cos(\theta)}$. So all you do is take the inverse which is $cot(\theta) = \frac{cos(\theta)}{sin(\theta)}$.

Plug values in

$cot(\theta) = \frac{-\frac{3}{4}}{\frac{4\sqrt{7}}{7}}$

Keep, change, flip

$-\frac{3}{4} * \frac{7}{4\sqrt7}$

Multiply:

$-\frac{21}{16\sqrt7}$

Multiply both sides by sqrt(7)

$-\frac{21\sqrt7}{112}$

This is the value of cot(theta)