QuestionFind the general solution for $\sin (x-30)=\cos 2x$ .

Studdy Solution

STEP 1

Assumptions1. We are looking for the general solution to the trigonometric equation $\sin (x-30)=\cos x$ . . We will use the identity \cosx =1 -\sin^ x to transform the equation into a form that can be solved more easily.

STEP 2

First, replace $\cos2x$ in the equation with the identity $1 -2\sin^2 x$ .
$\sin (x-30)=1 -2\sin^2 x$

STEP 3

Rearrange the equation to isolate the $\sin^2 x$ term.
$2\sin^2 x + \sin (x-30) -1 =0$

STEP 4

Now, we have a quadratic equation in terms of $\sin x$ . Let's make a substitution to simplify the equation. Let $y = \sin x$ .
$2y^2 + y\sin30 -1 =0$

STEP 5

We know that $\sin30 =0.5$ , so we can substitute this value into the equation.
$2y^2 +0.5y -1 =0$

STEP 6

Now, we have a quadratic equation in $y$ . We can solve this equation using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ .
$y = \frac{-0.5 \pm \sqrt{(0.5)^2 -42(-1)}}{2*2}$

STEP 7

olve for $y$ .
$y = \frac{-0.5 \pm \sqrt{0.25 +}}{4}$

STEP 8

implify the equation.
$y = \frac{-0.5 \pm \sqrt{8.25}}{4}$

STEP 9

Calculate the values for $y$ .
$y = \frac{-.5 \pm2.87}{4}$

STEP 10

We get two solutions for $y$ .
$y = \frac{-0.5 +2.87}{4} =0.59$ $y2 = \frac{-0.5 -2.87}{4} = -0.84$

STEP 11

Remember that $y = \sin x$ , so we can write the solutions in terms of $x$ .
$\sin x =0.59$ $\sin x = -0.84$

STEP 12

Now, we can find the general solutions for $x$ using the inverse sine function.
$x = \arcsin(0.59) +2n\pi$ $x = \pi - \arcsin(0.84) +2n\pi$ where $n$ is an integer.

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