Trigonometric Functions

6.6b Solving Trigonometric Equation (Factorization)   Example: Find all the angles that satisfy each of the following equations for 0° £ x £ 360°. (a)  cot x = − 2 cos x (b)  3 sec x = 4 cos x   (c)  16 tan x = cot x Solution: (a) cot x = − 2 cos x cos … Read more

6.6 Simple Trigonometric Equations Steps to solve simple trigonometric equation: (1) Determine the range of values of the required angles. (2) Find a basic angle by using calculator. (3) Determine the quadrants the angle should be. (4) Determine the values of angles in those quadrants. (A) Solving Trigonometric Equation (Basic Equation in sinx/ cosx/ tanx/ … Read more

Example 2: Prove each of the following trigonometric identities. (a) 1 + cos 2 x sin 2 x = cot x (b) cot A sec 2 A = cot A + tan 2 A (c) s i n x 1 − c o s x = cot x 2 Solution: (a) L H S = … Read more

6.5 Formulae of sin (A ± B), cos (A ± B), tan (A ± B), sin 2A, cos 2A, tan 2A (A) Compound Angles Formulae: • sin ( A ± B ) = sin A cos B ± cos A sin B • cos ( A ± B ) = cos A cos B ∓ … Read more

6.4 Basic Trigonometric Identities Three basic trigonometric identities are: sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x cot2 x + 1 = cosec2 x Example 1 (To Prove Trigonometric Identities which involve the Three Basic Identities) Prove each of the following trigonometric identities. (a) sin2 x – cos2 x = 1 – 2 cos2 x (b) (1 – cosec2 x) (1– … Read more

6.3.3 Sketching Graphs of Trigonometric Functions (Part 2) Example 2: (a) Sketch the graph y = –½ cos x for 0 ≤ x ≤ 2π. (b) Hence, using the same axes, sketch a suitable graph to find the number of solutions to the equation π 2 x + cos x = 0 for 0 ≤ x ≤ 2π. State the … Read more

6.3.2 Sketching Graphs of Trigonometric Functions Example 1: Sketch the graph of each of the following trigonometric functions for π ≤ x ≤ 2π. (a) y = 3 sin x (b) y = 2 cos x (c) y = sin x + 1 (d) y = cos x –1   (e) y = sin 2x    (f) y = cos 2x (g) … Read more

6.3.1 General pattern for y = sin x , y = cos x and y = tan x (1) Graph of y = sin x x 0o 90o 180o 270o 360o sin x 0 1 0 -1 0 (2) Graph of y = cos x x 0o 90o 180o 270o 360o cos x 1 0 -1 0 1 (3) Graph of y = tan x … Read more

6.2b Six Trigonometric Functions of Any Angle  (B) Special Angles (1) Value of Special Angle 30° and 60°     (a)sin 30 o = 1 2  (b)cos 30 o = 3 2 (c)tan 30 o = 1 3   (d)sin 60 o = 3 2   (e)cos 60 o = 1 2    (f)tan 60 o = 3   (2) … Read more

6.2a Six Trigonometric Functions of Any Angle  (A) The definition of sin, cos, tan, cosec, sec and cot 1. Let P (x, y) be any point on the circumference of the circle with centre O and of radius r. Based on ∆ OPQ in the above diagram, 2. The definitions of tangent, cotangent, secant and cosecant of any angle are: 3. The … Read more