Chapter 11 Trigonometric Functions 11.1 Trigonometric Ratios and General Angles 11.2 Trigonometric Ratios of Any Angles 11.3 Graphs of Sine, Cosine and Tangent Functions

Trigonometric Equations Objectives 11.1 Trigonometric Ratios and General Angles In this lesson, we will learn how to find the trigonometric ratios for acute angles, particularly those for 30°, 45° and 60° (or respectively in radians).

Trigonometric Ratios of Acute Angles The three trigonometric ratios are defined as OPQ is a right angled triangle Trigonometric Equations adjacent opposite hypotenuse opposite hypotenuse adjacent

Example 1 In the right-angled triangle ABC, tan θ = 2. Find sin θ and cos θ. Solution Trigonometric Equations A B C θ Since tan θ =, 2 1 BC = 2 units and AB = 1 unit. By Pythagoras’ Theorem, AC =.

Trigonometric Ratios of Special Angles Draw a diagonal to the square. Draw a unit square. Trigonometric Equations The length of the diagonal is √2 and the angle is 45°.

Trigonometric Ratios of Special Angles Draw an equilateral triangle of side 2 cm. Trigonometric Equations The altitude bisects the base of the triangle. Draw an altitude. The length of the altitude is √3 and the angles are 60° and 30°.

Trigonometric Ratios of Complementary Angles In the right-angled triangle OPQ but  OPQ = 90°– θ Trigonometric Equations  OPQ = – θ If θ is in radians

Trigonometric Equations Example 2 Using the right-angled triangle in the diagram, show that sin(90 0 – θ) = cos θ. Hence, deduce the value of Solution P Q R θ a b c 90 0 – θ Thus, sin(90 0 – θ) = cos θ. sin 70 0 = sin (90 0 – 20 0 ) = cos 20 0

Trigonometric Equations Consider angles in the Cartesian plane. OP is rotated in an anticlockwise direction around the origin O. The basic (reference) angle that OP makes with the positive x–axis is α. Now OP is rotated in the clockwise direction. 1st quadrant 2nd quadrant 4th quadrant 3rd quadrant

Example 3 Given that 0 0 < θ < and the basic angle for θ is 40 0, find the value of θ if it lies in the (a) 3 rd quadrant,(b) 4 th quadrant. Solution (a)(b) Trigonometric Equations

Using the complementary angle identity. Substitute for sin θ. Trigonometric Equations Using the complementar y angle identity. Substitute for tan A. Exercise 11.1, qn 2

Trigonometric Equations Exercise 11.1, qn 5(a) Solution

Trigonometric Equations Exercise 11.1, qn 5(c) Solution

Trigonometric Equations Find all the angles between 0° and 360° which make a basic angle of 70°. The angles are as follows: Exercise 11.1, qn 7(b) Solution

Find all the angles between 0 and 2π which make a basic angle of The angles are as follows: Exercise 11.1, qn 7(b) Solution Trigonometric Equations