Question Solve the trigonometric equation $\cot^2 x + 12 \cot x = 0$ for $x$ .

Studdy Solution

STEP 1

Assumptions
1. The equation is $\cot^2 x + 12 \cot x = 0$ .
2. We need to find the values of $x$ that satisfy the equation.

STEP 2

Recognize that the equation is a quadratic in terms of $\cot x$ . Let's set $u = \cot x$ to simplify the equation.
$u^2 + 12u = 0$

STEP 3

Factor the quadratic equation.
$u(u + 12) = 0$

STEP 4

Set each factor equal to zero and solve for $u$ .
$u = 0 \quad \text{or} \quad u + 12 = 0$

STEP 5

Solve the first equation for $u$ .
$u = 0$

STEP 6

Solve the second equation for $u$ .
$u + 12 = 0 \Rightarrow u = -12$

STEP 7

Now we need to translate the solutions for $u$ back into solutions for $x$ using the original substitution $u = \cot x$ .
For $u = 0$ :
$\cot x = 0$

STEP 8

Solve for $x$ when $\cot x = 0$ . Recall that $\cot x = \frac{\cos x}{\sin x}$ , and this ratio is zero when $\cos x = 0$ and $\sin x \neq 0$ .

STEP 9

Find the values of $x$ where $\cos x = 0$ and $\sin x \neq 0$ . This occurs at the points where $x$ is an odd multiple of $\frac{\pi}{2}$ .
$x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$

STEP 10

For $u = -12$ :
$\cot x = -12$

STEP 11

Solve for $x$ when $\cot x = -12$ . We need to find the angle whose cotangent is -12. This will be an angle in the second or fourth quadrant, where the cotangent is negative.

STEP 12

Use the definition of cotangent, $\cot x = \frac{1}{\tan x}$ , to find the angle whose tangent has the value $-\frac{1}{12}$ .
$\tan x = -\frac{1}{12}$

STEP 13

Find the principal value of $x$ where $\tan x = -\frac{1}{12}$ . We can use an inverse tangent function to find this value.
$x = \arctan\left(-\frac{1}{12}\right)$

STEP 14

Since the tangent function has a period of $\pi$ , we can find all solutions for $x$ by adding integer multiples of $\pi$ to the principal value.
$x = \arctan\left(-\frac{1}{12}\right) + k\pi, \quad k \in \mathbb{Z}$

STEP 15

Combine the solutions from steps 9 and 14 to get the complete set of solutions for the original equation.
$x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$ $x = \arctan\left(-\frac{1}{12}\right) + k\pi, \quad k \in \mathbb{Z}$
These are the values of $x$ that satisfy the equation $\cot^2 x + 12 \cot x = 0$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms