Find all real numbers in the interval 02 pi that satisfy the

Find all real numbers in the interval (0,2 pi) that satisfy the equation. 6 sec^2 theta tan theta = 8 tan theta The solution set is{}. (Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) Find all values of theta in the interval [0 degree, 360 degree) that satisfy the equation. Use degree measure. 2sin^theta = cos^theta The solution set is{}. (Round to the nearest tenth as needed. Do not include the degree symbol in your answer. Use a comma to separate answers as needed.) Find all real numbers on the interval [0, 2 pi] that satisfy the equation. Use radian measure. 2sin^2x - 3sin x = -1 The solution set is (Simplify your answer. Type an exact answer, using pi as needed. Use a comma to separate answers as needed.) Find all real numbers in the interval [0, 2 pi] that satisfy the equation. Use radian measure. 9sin^2x = sin x The solution set is{}. (Round to the nearest tenth as needed. Use a comma to separate answers as needed.)

Solution

1 ) 6 sec^2 theta tan theta = 8 tan theta

subtracting 8 tan theta from both sides

6 sec^2 theta tan theta - 8 tan theta = 0

taking gcf as 2 tan theta

2 tan theta ( 3 sec^2 theta - 4 ) = 0

2 tan theta ( sqrt 3 sec theta - 2) ( sqrt 3 sec theta + 2) = 0

tan theta = 0 , theta = 0 ,pi,2pi

or sqrt 3 sec theta - 2 = 0 , theta = 5pi/6 , 7pi/6

or sqrt 3 sec theta + 2 = 0 , theta = pi/6 , 11 pi/6

combining all solutions

theta = 0, pi/6 , 5pi/6 , 7pi/6,11pi/6 ,pi,2pi

2) 2 sin theta = cos theta

dividing both sides by cos theta

2 sin theta / cos theta = cos theta / cos theta

2 tan theta = 1

tan theta = 1/2

theta = tan^-1 ( 1/2 )

solutions are

theta = pi + .4636 , theta = .4636

3) 2 sin^2 x - 3 sin x = -1

add 1 to both sides

2 sin^2 x - 3 sin x + 1 =0

( sin x -1 ) ( 2 sin x -1 ) = 0

sin x = 1 , x = pi/2

, sin x = 1/2 , x = pi/6 , 5pi/6

combining the solutions

x = pi/2 , pi/6 , 5pi/6

factoring the equation