Question

Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0, 21). Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution." - 20cos(x) – 3cos(x) = -2 Answer How to enter your answer Keypad Keyboard Shortcuts Enter your answer in radians, as an exact answer when possible. Multiple solutions should be separated by commas. Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answer is used. x = O No Solution

Answer 1

-20cos (2) - 3cos(x) = -2

$\small -20cos^{2}(x)-3cos(x)+2=0$

$\small 20cos^{2}(x)+3cos(x)-2=0$

$\small 20cos^{2}(x)+8cos(x)-5cos(x)-2=0$

$\small 4cos(x)(5cos(x)+2)-1(5cos(x)+2)=0$

(4cos(2) - 1) (5cos(x) + 2) = 0

$\small cos(x)=\frac{1}{4},-\frac{2}{5}$

$\small \textup{i) }cos(x)=\frac{1}{4}$

$\small \Rightarrow x=cos^{-1}\left ( \frac{1}{4} \right )=1.3181,4.9651$

$\small \textup{ii) }cos(x)=-\frac{2}{5}$

$\small \Rightarrow x=cos^{-1}\left (- \frac{2}{5} \right )=1.9823,4.3009$

$\small \boldsymbol{\therefore x=1.3181,1.9823,4.3009,4.9651}$