M3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.

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1 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides of sin θ cos θ + = to see another form of this identity. b. Subtract sin θ from both sides of sin θ cos θ + = to see another form of this identity.. There are two more variations of the Pythagorean Identity, involving the other trigonometric functions. a. To find one of the variations, divide both sides of simplify each term. sin θ cos θ + = by cos θ, and b. Subtract from both sides of your equation in part (a) to write this identity in a different form. c. To find another variation of the Pythagorean Identity, divide both sides of sin θ + cos θ = by sin θ, and simplify each term. d. Subtract from both sides of your equation in part (c) to write this identity in a different form.

2 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 Part 3: Simplifying Trigonometric Epressions General guidelines for simplifying trig epressions: More simplified means fewer operations and fewer fractions. Positive angles are simpler than negative angles. For eample, tan is more simplified tan. than ( ) Things to try: Substitute one side of a trig identity in place of the other side. For eample, replaced by ( sin θ ). cos θ can be Factor or multiply. Use a common denominator to add or subtract fractions. Write all parts in terms of sines and cosines. When you see sums or differences of squares and s, substitute with a Pythagorean identity. 5. Before we begin simplifying, let s write all of our trig identities together in one place: a. Reciprocal identities: cscθ = secθ = cotθ = b. Quotient identities: tanθ = cotθ = c. Pythagorean identities: (see page ) d. Cofunction identities: (see page of packet from unit 3) π sin θ = π cos θ = π tan θ = π csc θ = π sec θ = π cot θ = e. Even/odd identities: sin csc ( θ ) = cos ( θ ) = tan ( θ ) ( θ ) = sec ( θ ) = cot ( θ ) = =

3 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Simplify these trigonometric epressions. A hint is given for each. a. csc tan Write in terms of sines and cosines. b. + cot cos Write in terms of sines and cosines, and get a common denominator. c. ( secθ )( secθ ) + Multiply. Then use a Pythagorean identity to substitute. d. + cot sec Use a Pythagorean identity, or write in terms of sines and cosines. e. + cos + cos Get a common denominator, and use a Pythagorean identity to substitute. f. sin cos ( ) ( ) Use even/odd identities. 3

4 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO VERIFYING TRIGONOMETRIC IDENTITIES 5. Part : The Basics A trigonometric identity is a statement about trig functions that is true for all values in their domain. For eample, the identity tan cos To verify an identity, use algebra to prove the two sides equal. = is true for all -values at which tan eists. Because you are proving equality, you are not allowed to use properties of equality in your proof. Thus, adding, subtracting, multiplying, or dividing the same number to both sides is not allowed. In fact, you cannot do anything that changes the value of either side, such as squaring, taking the square root, etc You must either work with one side until you get it to match the other, or work on each side separately until you get them to match.. One technique is to work with only the more complicated-looking side. Try simplifying the right side until you get it to match the left side: cotθ = cosθcscθ. If you see a sum or difference of fractions, use a common denominator to combine the fractions first. Try this one, only working on the left side: cos cos + = + cos 4

5 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Try simplifying the left side by multiplying: ( ) sinθ + cosθ tanθ = tanθ + sin θ 4. As with simplifying, when you see sums or differences of squares and s, it may be helpful to substitute with Pythagorean identities on one or both sides: = sec Part : More Advanced Techniques 5. This identity is already written in terms of sine and cosine and does not contain squares. By our guidelines on page, you d be stuck! A more advanced technique is to multiply the numerator and denominator by a conjugate to create a difference of squares. Multiply : the numerator and denominator of the left side by ( + ), the conjugate of ( ) cos + = cos 5

6 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Factoring can also be helpful. Factor the numerator of the left side: = cos 4 sin cos 7. To simplify a comple fraction (fractions within a fraction), multiply the numerator and denominator by an LCD to eliminate denominators within the fraction. Here, convert the left side to sines and cosines, and simplify the comple fraction: cos tan + cot = 8. Look for features that are easy to match. In this eample, the left side has sec in the numerator, and the right side has cos in the denominator. Make one of these match the other, and then see what you can do with the rest: ( sec )( sec ) + = + cos sec tan tan 6

7 5D M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 PRACTICE VERIFYING TRIGONOMETRIC IDENTITIES 5D Verify each trigonometric identity. Use the hints on p. 4-6 of this packet to help you start.. csc = csc csc. sec β + tan β = tan β tan β 3. cos y sec y+ tan y = sin y 4. cos sin tan = cos 5. cos + = 0 cos + 7

8 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO SOLVING TRIGONOMETRIC EQUATIONS 5.3 Part : Finding solutions in [ ) 0,π. Label the radian measures of the angles in the other 3 quadrants with the given reference angle. π π 3 π 4 6. To solve a trig equation, use inverse operations to isolate the trig function. a. In the equation + 3 = 0, first isolate. b. Now list all angles in [ 0,π ) that satisfy the resulting statement. 3. Sometimes, you need to factor to isolate the trig function. a. Factor the equation 0 =. (If you can t factor as is, let u sin = first.) b. Set each factor equal to 0, and isolate in each. c. List all angles in [ 0,π ) that satisfy the resulting statements. 8

9 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO Do you see more than one trig function? Use an identity to rewrite in terms of only one. Here, convert to all cosines, and then find all solutions in [ 0,π ) : 3cos + 3 = sin 5. In this equation, remember the ± after you take the square root of both sides: Find all solutions in [ 0,π ) for 4cos 3 = 0. Part : Finding a General Formula for Solutions 6. So far, we ve only found solutions in the interval [ 0,π ), but all of these equations really have infinite solutions. Why is that? 7. a. Isolate in + = 0. b. In the interval [ 0,π ), there are solutions. Fill them in the blanks below: = + nπ or = + nπ, where n is any integer The + nπ makes this a general solution. It means that you can take the angle measures you found and add π ( ), π ( ), ( ) makes the equation true. 3π, etc to find another angle that For sine and cosine equations, we use + nπ in our general formulas because the period of the graph is π. 9

10 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO a. Isolate tan in tan =, and fill in the blanks with solutions from the interval [ ) 3 0,π : = + nπ or = + nπ, where n is any integer b. The + nπ means that we can add π, π, 3π, etc to find another angle that solves the equation. Why do we use nπ here instead of nπ? 9. Solve these, giving general formulas for your answers. a. + = b. = 3( cos ) cos cos 0 Part 3: Solving Equations with b 0. Suppose cos = : a. Fill in the blanks with angles in [ 0,π ) that have cosine value = + nπ or = + nπ, where n is any integer : b. Divide both sides of each equation by to finish solving for. c. Now use your formulas from part (b) to find all angles in the interval [ 0,π ) that solve the original equation. 0

11 . Consider the equation ( ) a. Isolate sin( 3 ). M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 sin 3 = 0. b. Fill in the blanks with angle measures in [ 0,π ) that have the sine value shown in your isolated equation: 3 = + nπ or 3 = + nπ or 3 = + nπ or 3 = + nπ, n is any integer c. Divide by 3 to finish solving for. d. Use your formulas to list all angles in [ 0,π ) that solve the original equation.. Find a general formula and all solutions in [ 0, π ) for the equation ( ) tan =. 3. Find a general formula and all solutions in [ 0,π ) for the equation 3 sin =.

12 Review M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 REVIEW OF 5. TO 5.3 Review Identities to memorize: = csc csc = cos = sec sec = cos tan = cot cot = tan tan = cos cos cot = Pythagorean Identity: + cos = π Cofunction Identities: sin = cos π cos = π csc = sec π sec = csc cos cos Odd/Even Identities: ( ) = ( ) = ( ) csc( ) = csc sec( ) = sec cot ( ) π tan = cot π cot = tan tan = tan = cot. Derive all other forms of the Pythagorean Identity, and write them below. Simplify.. sec + + tan cos cos cos cos cos tan tan + sec sec + 6. cot cos cos + 7. ( sec + tan )( sec tan )

13 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 Verify each identity. 8. tan sec cos = 9. ( )( ) sec + tan = cos sec tan = ( ) sin + cos + = csc + cos. csc + sec = csc sec 3. cos cos + = sec csc cos 3

14 M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 Find a general formula and all solutions in [ 0,π ). 4. cos = = 6. sin 3sin 0 3sec = sec Find a general formula and all solutions in [ 0,π ). 7. sin = 3cos 8. cos = 9. sin3 = 0. tan 4 =. 4 tan sec + 3 = 0 Answers:. csc 3. cos 4. cot 5. csc 6. tan = nπ; { 0, π} π 5π π π 5π π 5. = + nπ, + nπ, + nπ;,, , 6. no solution π 5π π 5π 7. = + nπ, + nπ;, π π 3π 8. nπ ;, π nπ 5π nπ π 3π 5π 5π 7π 9π = + 9. = +, + ;,,,,, π nπ 3π 7π π 5π 9π 3π 7π 3π 0. = + ;,,,,,,, π nπ π 3π 5π 7π. = + ;,,,