Trigonometric Functions Long Questions (Question 1 - 3)


Question 1:
(a) Sketch the graph of y = cos 2x for 0°  x  180°.
(b) Hence, by drawing a suitable straight line on the same axes, find the number of 
      solutions satisfying the equation 2  sin 2 x=2 x 180 for 0°  x  180°.

Solution:
(a)(b)

2  sin 2 x=2 x 180 12  sin 2 x=1( 2 x 180 ) cos2x= x 180 1 y= x 180 1 x=0,  y=1 x=180,  y=0 Number of solutions = 2


Question 2:
(a) Sketch the graph of   y= 3 2 cos2x for 0x 3 2 π
(b) Hence, using the same axes, sketch a suitable straight line to find the number of
      solutions to the equation 4 3π xcos2x= 3 2  for 0x 3 2 π
      State the number of solutions.

Solution:
(a)(b)
4 3π xcos2x= 3 2 cos2x= 4 3π x 3 2 3 2 cos2x= 3 2 ( 4 3π x 3 2 ) y= 2 π x 9 4 To sketch the graph of y= 2 π x 9 4 x=0, y= 9 4 x= 3π 2 , y= 3 4 Number of solutions  =Number of intersection points = 3



Question 3:
(a) Prove that 2tanx 2 sec 2 x =tan2x. .                                                         
(b)(i) Sketch the graph of y = – tan 2x for 0  x ≤ p .    
(b)(ii) Hence, by drawing a suitable straight line on the same axes, find the number of solutions satisfying the equation 3x π + 2tanx 2 sec 2 x =0  for 0  x  p .
State the number of solutions.

Solution:
(a)
2tanx 2 sec 2 x =tan2x LHS: 2tanx 2 sec 2 x = 2tanx 2( 1+ tan 2 x )             = 2tanx 2 tan 2 x             =tan2x (RHS)
(b)(i) 

(b)(ii)
3x π + 2tanx 2 sec 2 x =0 3x π +tan2x=0 from part (a) tan2x= 3x π              y= 3x π The suitable straight line to sketch is y= 3x π .
When x = 0, y = 0.
When x = p, y = 3.
          Number of solutions = 3