Find the solutions on the given interval For a first use a c

Find the solutions on the given interval. For (a), first use a complementary angle identity in order to have only one trig function. Then use the sum-to-product identity, simplify and solve. 2tan(x) = tan^2(x) - 1 [0, pi)

Solution

2tanx = tan^2x -1

tan^2x -2tanx -1 =0

solve the quadratic using formula:

tanx = ( 2 + /- sqrt( 4 +4) )/2

= ( 2 +/- 2sqrt2)/2

= 1+/- sqrt2

tanx = 1+sqrt2 = 2.41

x = tan^-1(2.41) = 1.17 rad

; tanx = 1 -sqrt2 = -0.41

x = tan^-1(-0.41) = 2.75

In the interval [ 0, pi ) x = 1.17 , 2.75 rad