6.8.1 Trigonometric Functions Long Questions (Question 1 & 2)


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

Solution:
(a)
2 tan x 2 sec 2 x = tan 2 x L H S : 2 tan x 2 sec 2 x = 2 tan x 2 ( 1 + tan 2 x ) = 2 tan x 2 tan 2 x = tan 2 x (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 = π, = 3.
  Number of solutions = 3

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