5.6b Solving Trigonometric Equation (Factorization)

5.6b Solving Trigonometric Equation (Factorization)

Example:
Find all the angles that satisfy each of the following equations for £ x £ 360°.
(a)  cotx=2cosx
(b)  3secx=4cosx   
(c)  16tanx=cotx

Solution:
(a)
cotx=2cosx cosx sinx =2cosx cosx=2cosxsinx cosx+2sinxcosx=0 cosx( 1+2sinx )=0 cosx=0 x= 90 , 270 1+2sinx=0 sinx= 1 2 Basic=3 0 x=( 180 + 30 ),( 360 30 ) x= 210 , 330 x= 90 , 210 , 270 , 330

(b)
3secx=4cosx 3 cosx =4cosx 3=4 cos 2 x cos 2 x= 3 4 cosx=± 3 2 Basic = 30 x= 30 ,( 180 30 ),( 180 + 30 ),( 360 30 ) x= 30 , 150 , 210 , 330

(c)
16tanx=cotx 16tanx= 1 tanx tan 2 x= 1 16 tanx=± 1 4 Basic  = 14.04 x= 14.04 ,( 180 14.04 ),      ( 180 + 14.04 ),( 360 14.04 ) x= 14.04 , 165.96 , 194.04 , 345.96