TRIG. IDENTITIES ANALYTICAL TRIGONOMETRY 5.2 & 5.3

Transcription

1 TRIG. IDENTITIES ANALYTICAL TRIGONOMETRY 5.2 & 5.3 TODAY S AGENDA Answer any questions that you may have about the problems from the book. Look at 5.2 and 5.3. Objective: 5.2 you learned how to verify trigonometric identities. Objective: 5.3 you learned how to use standard algebraic techniques and inverse trigonometric functions to solve trigonometric equations. Find steps for success when dealing with identities. Go over problems that may occur in your homework. Be done with new material! 1

2 VERIFYING TRIGONOMETRIC IDENTITIES This can be difficult so make sure that you take time to do your best. There are no set techniques that you apply every time when verifying identities. The key to verifying identities is; the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions. An identity is an equation that is true for all real values in the domain of the variable. GUIDELINES 1. Work with only one side of the equation at a time. Usually it is better to start with the more complicated side first. 2. Look for opportunities to factor an expression, add fractions, square a two term quantity, or create a single term denominator. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sine and cosine pair well, as do secant and tangent, and cosecant and cotangents. 4. As a last resort, convert all terms to sine and cosine. 5. Always try something! Even paths that lead to dead ends give you insight. 2

3 NOW FOR THE EXAMPLES: VERIFY THE IDENTITY sin tan cos sec Begin by converting all terms to sines and cosines. VERIFY THE IDENTITY sin csc 1 csc 1 1 sin Because the left side is more complicated, start with it. Begin by multiplying (csc x - 1) by (csc x + 1), and then search for a fundamental identity that can be used to replace the result. 3

4 VERIFY THE IDENTITY cot = cot csc cot. VERIFY THE IDENTITY = 2 cot 4

5 VERIFY THE IDENTITY (1 + cot )(1 sin ) = cot VERIFY THE IDENTITY sec + tan = 5

6 TRY THIS cot cos = csc sin VERIFY THE IDENTITY = 6

7 5.3 SOLVING TRIGONOMETRIC EQUATIONS Section Objectives: Students will know how to use standard algebraic techniques and inverse trigonometric functions to solve trigonometric equations. IMPORTANT: your preliminary goal in solving trigonometric equations is to isolate the trigonometric function involved in the equation. We will be using many different techniques to solve trig. equations. Make sure you have your thinking caps on and ready to work. You will also need your unit circle available and p.376. THINK ABOUT THIS! How many solutions does the equation sec x = 2 have? Explain. The equation has an infinite number of solutions because the secant function has a period of 2. Any angles coterminal with the equation s solutions on [0, 2 ) will also be solutions of the equation. What are those solutions then? 7

8 EXAMPLE #1 Solve 1 2 cos = cos = 0 cos = 1/2 = = EXAMPLE #2 Solve sin + 1 = sin. sin + 1 = sin 2sin + 1 = 0 sin = 1 2 = =

9 EXAMPLE #3 sec csc = csc sec csc csc = 0 csc (sec 1) = 0 csc = 0 sec 1 = 0 sec = 1 = 2 EXAMPLE OYO Solve 1) 2 cos 1 = 0 2) tan 3 = 0 = +, = + x = +, = + 9

10 EQUATIONS OF QUADRATIC TYPE To solve a trigonometric equation of quadratic type, factor the quadratic, or if factoring is not possible, use the Quadratic Formula. In the questions, we may have a restriction, such as [0,2 ) Which is within the unit circle. If there are no restrictions, we us the period that they always exist in, such as +. Care must be taken when squaring both sides of a trigonometric equation to obtain a quadratic because this procedure can introduce extraneous solutions, so any solutions must be checked in the original equation to see whether they are valid or extraneous. EXAMPLE #4 Solve the following on the interval [0, 2 ) 2cos + cos 1 = 0 (2cos 1)(cos + 1) = 0 2cos 1 = 0 cos + 1 = 0 cos = 1 2 cos = 1 = 3, 5 3 = 10

11 EXAMPLE #5 Solve the following on the interval [0, 2 ) 2cos + 3sin 3 = 0 2(1 sin ) + 3sin 3 = 0 2sin 3sin + 1 = 0 (2sin 1)(sin 1) = 0 2sin 1 = 0 sin 1 = 0 sin = 1 2 sin = 1 = 6, 5 6, 2 EXAMPLE OYO Solve tan + 2 tan = 1. 11

12 EXAMPLE OYO: SOLVE sec + 1 = tan FUNCTIONS INVOLVING MULTIPLE ANGLES Solve the following on the interval [0, 2 ). 2 sin = 0 sin 2 = 1/2 2 = 4 3, 5 3, 10 3, 11 3 = 2 3, 5 6, 5 3, 11 6 Tip: Note that since 0 2,

13 ANOTHER EXAMPLE OYO cot = 0 Tip: Note that since 0 2, 0 OYO sin 4 = 13

14 HOMEWORK p.389 #1-9odd, 23-55eoo p.400 #1,13-39eoo 14