QuestionFind $\csc \theta$ if $\cot \theta = -\frac{1}{6}$ and $\theta$ is in quadrant IV.

Studdy Solution

STEP 1

Assumptions1. $\cot \theta=-\frac{1}{6}$ . $\theta$ is in quadrant IV3. We need to find the exact value of $\csc \theta$

STEP 2

In any right triangle, the cotangent of an angle is the ratio of the adjacent side to the opposite side. Since $\cot \theta=-\frac{1}{6}$ , we can consider a right triangle where the adjacent side is -1 (negative because $\theta$ is in quadrant IV where x is positive and y is negative) and the opposite side is6.

STEP 3

We can use the Pythagorean theorem to find the hypotenuse of the triangle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
$Hypotenuse^2 = Adjacent^2 + Opposite^2$

STEP 4

Plug in the values for the adjacent and opposite sides to calculate the hypotenuse.
$Hypotenuse^2 = (-1)^2 +6^2$

STEP 5

Calculate the square of the hypotenuse.
$Hypotenuse^2 =1 +36 =37$

STEP 6

Take the square root of both sides to find the length of the hypotenuse.
$Hypotenuse = \sqrt{37}$

STEP 7

The cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the opposite side. So, we can calculate $\csc \theta$ as follows $\csc \theta = \frac{Hypotenuse}{Opposite}$

STEP 8

Plug in the values for the hypotenuse and the opposite side to calculate $\csc \theta$ .
$\csc \theta = \frac{\sqrt{37}}{6}$ Therefore, the exact value of $\csc \theta$ is $\frac{\sqrt{37}}{6}$ .

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