Trigonometric Functions

Question 3: (a) Sketch the graph of y= 3 2 cos2x for 0≤x≤ 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 x−cos2x= 3 2  for 0≤x≤ 3 2 π State the number of solutions. Solution: (a)(b) 4 3π x−cos2x= 3 2 … Read more

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 1−2  sin 2 x=1−( 2− x … Read more

Question 15: Prove the identity 2 cos 2 A + 1 = s e c 2 A Solution: LHS = 2 cos 2 A + 1 = 2 ( 2 cos 2 A − 1 ) + 1 ← cos 2 A = 2 cos 2 A − 1 = 2 2 cos 2 A … Read more

Question 11: Prove the identity cos 2 x 1−sinx =1+sinxSolution: LHS = cos 2 x 1−sinx = 1− sin 2 x 1−sinx ← sin 2 x+ cos 2 x=1 = ( 1+sinx )( 1−sinx ) 1−sinx =1+sinx =RHS ∴Proven Question 12: Prove the identity sin 2 x− cos 2 x= tan 2 x−1 tan 2 … Read more

Question 9: Given that sin θ = 3 5 , where θ is an acute angle, without using tables or a calculator, find the values of (a) sin (180º + θ), (b) cos (180º – θ), (c) tan (360º + θ). Solution: (a) sin θ = 3 5 cos θ = 4 5 tan θ … Read more

Question 7: It is given that   sin A = 5 13 and cos B = 4 5 , where A is an obtuse angle and B is an acute angle. Find (a) tan A (b) sin (A + B) (c) cos (A – B)    Solution: (a) tan A = − 5 12 (b) sin … Read more

Question 5: Find all the angles between 0° and 360° which satisfy the following equations: (a) 2 sin ( 2x – 50o) = –1  (b) 15 sin2x = sin x + 4 sin 30o (c) 7 sin x cos x = 1   Solution: (a) 2 sin ( 2x – 50o) = –1  sin ( 2x – 50o) = – … Read more

Question 1: Solve the equation 3 cos 2A = 8 sin A – 5 for 0° ≤ A ≤ 360°. Solution: 3 cos 2A = 8 sin A – 5 3(1–2 sin2 A) = 8 sin A – 5 3 – 6 sin2 A = 8 sin A – 5 6 sin2 A + 8 sin A – 8 … Read more

(D) Solving Trigonometric Equations (Involving Addition Formulae and Double Angle Formulae) Example 1 (Addition Formulae): Solve the following equation for 0o ≤ x ≤ 360o: (a) sin ( x – 25o) = 3 sin ( x + 25o) (b) 3 cos ( 2x + 10o) = 2    Solution:   (a) sin ( x – 25o) = 3 sin … Read more

(C) Solving Trigonometric Equation (Form Quadratic Equation in sinx/ cosx/ tanx/ cosecx/ secx/ cotx) Example: Find all the angles between 0° and 360° that satisfy each of the following equations. (a) 3 sin² x – 2 sin x – 1 = 0 (b)  2 sin x = cosec x + 1 (c) 5 sin² x = 2 (1 + cos x) … Read more