Analytic Trigonometry PDF - University of Cyberjaya

ANALYTICS TRIGONOMETRY Basic Mathematics (FADS 1004) © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. CHAPTER 6: ANALYTIC TRIGONOMETRY 1.Trigonometric Identities...

ANALYTICS TRIGONOMETRY Basic Mathematics (FADS 1004) © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. CHAPTER 6: ANALYTIC TRIGONOMETRY 1.Trigonometric Identities 2.Trigonometric Equations 3.Sum and Difference Formula © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author.  Trigonometric Identities Learning Objectives: At the end of this lesson, student should be able to: 1. Recall the basic trigonometric identities 2. Verify other trigonometric identities by using basic trigonometric identities © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. Trigonometric Identities © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author.  Example: Trigonometric Identities Simplify the following identities: cot 𝜃 1) csc 𝜃 (1− cos2 𝑥) 2) 𝑠𝑖𝑛𝑥 © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author.  Example: Trigonometric Identities Prove the following trigonometry identity: 1−sin 𝜃 cos 𝜃 1. cos 𝜃 = 1+sin 𝜃 2 3 2. cos 𝑥 − cos 𝑥 sin 𝑥 = cos 𝑥 3. sin 𝑥 + cos 𝑥 cot 𝑥 = csc 𝑥 © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author.  Trigonometric Identities Learning Objectives: At the end of this lesson, student should be able to: 1. Solve trigonometric equations involving simple angles. © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author.  Example: 1 Find the solution of the equation of sin 𝜃 =. 2 a) 𝜃 is in the interval [0,2𝜋] b) 𝜃 is any real number © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. 𝑏) 𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝜋 1. 𝜃 = + 2𝜋𝑛 𝑛 = 0, 1, 2, 3, … 6 5 2. 𝜃 = 𝜋 + 2𝜋𝑛 6 © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author.  Example: Solve for 𝜃 for the following trigonometric equations. 2 1. 2 cos 𝜃 + 3 cos 𝜃 + 1 = 0 © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. Example: 1 −1 1 2 cos 𝜃 = − 𝜃′ = cos - Change to positive 2 cos 𝜃 + 3 cos 𝜃 + 1 = 0 2 2 𝜋 𝜃′ = 60°// 𝑙𝑒𝑡 cos 𝜃 = 𝑥 3 2 2 2𝑥 + 3𝑥 + 1 = 0 𝑄2 ∶ 𝜃 = 120°// 𝜋 3 4 (2𝑥 + 1)(𝑥 + 1) = 0 𝑄3 ∶ 𝜃 = 240°// 𝜋 3 −1 2𝑥 + 1 = 0 𝑥+1=0 cos 𝜃 = −1 𝜃′ = cos 1 - Change to positive ′ 1 𝜃 =0 𝑥=− 𝑥 = −1 2 𝑄2 ∶ 𝜃 = 180°// 𝜋 1 cos 𝜃 = − cos 𝜃 = −1 𝑄3 ∶ 𝜃 = 180°// 𝜋 2 2 4 𝐴𝑛𝑠 𝜃 = 120°, 240°, 180° // 𝜋, 𝜋, 𝜋 3 3 © 2023, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. 3. Sum and Difference Formulas Solve trigonometric equations involving Learning Objectives: sum and difference angles. 13 Sum and Difference Formulas Example: Express in term of a trigonometric function of 𝜃: (a) sin(θ − 3π 2 ) Example: (b) cos(θ + π) 4. Multiple Angle Formulas Learning Objectives: ✓Solve trigonometric equations involving double and half angle. ✓Use multiple-angle formulas to rewrite and evaluate trigonometric functions. 21 Multiple-Angle Formulas 23 Example – Functions of Multiple Angles Solve involving multiple angles. 2 cos 3𝑡 – 1 = 0. Solution: 2 cos 3𝑡 – 1 = 0 2 cos 3𝑡 = 1 24 Example 7 – Solution cont’d  5 In the interval [0, 2), you know that 3𝑡 = and 3𝑡 = are the only 3 3 solutions, so, in general, you have General solution Dividing these results by 3, you obtain the general solution, where n is an integer. Example: 1 Solve the equation for cos 2𝑥 = 2 25  Thank you Address Telephone Website University of Cyberjaya 03 - 8313 7000 www.cyberjaya.edu.my Persiaran Bestari, Cyber 11, 63000 Cyberjaya, Facsimile Email Selangor Darul Ehsan, Malaysia. 03 – 8313 7001 [email protected]