Term 3 : Unit 1 Trigonometric Functions Name : ____________ ( ) Class : _____ Date : _____ 1.1 Trigonometric Ratios and General Angles 1.2 Trigonometric Ratios of Any Angles

Trigonometric Equations Objectives 1.1 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. Example 4

Trigonometric Equations Example 5 Solution

Trigonometric Equations Find all the angles between 0° and 360° which make a basic angle of 70°. The angles are as follows: Example 6 Solution

Trigonometric Equations Objectives In this lesson, we will learn how to extend the definitions of sine, cosine and tangent to any angle, determine the sign of a trigonometric ratio of an angle in a quadrant, relate the trigonometric functions of any angle to that of its basic (reference) angle and solve simple trigonometric equations. 1.2 Trigonometric Ratios of Any Angles

Trigonometric Ratios of Any Angles The three trigonometric ratios are defined as Trigonometric Equations x y r yy r x r x PQ = y OQ = x

Example 7 Find the values of cos θ, sin θ and tan θ when θ = Solution Trigonometric Equations When θ = 135 0, – θ = 45 0.(basic angle) P has coordinates (1, -1) and

Trigonometric Equations

Signs of Trigonometric Ratios in Quadrants 1st quadrant Trigonometric Equations θ = α P has coordinates ( a, b ) 2nd quadrant θ = ( 180° – α ) P has coordinates ( – a, b ) 3rd quadrant θ = ( 180° + α ) P has coordinates ( – a, – b ) 4th quadrant θ = ( 360° – α ) P has coordinates ( a, – b ).

Trigonometric Equations For positive ratios Signs of Trigonometric Ratios in Quadrants In the four quadrants S (sin θ)A ( all ) T (tan θ) C (cos θ) The signs are summarised in this diagram.

Trigonometric Equations Example 8 Without using a calculator, evaluate cos 120°. Solution Basic angle, 120° is in the 2nd quadrant, so cosine is negative AS TC

Trigonometric Equations Example 9 Find all the values of θ between 0° and 360° such that sin θ = – 0.5. Solution For the basic angle, Since sin θ < 0, θ is in the 3rd or 4th quadrant, AS TC Basic Trigonometric Equations

Trigonometric Equations For the basic angle, θ is in the 1st, 2nd, 3rd or 4th quadrant, AS TC Example 10 Find all the values of θ between 0 o and 360 o such that 2sin 2 θ – 1 = 0. Solution