6.1: Verifying Trigonometric Identities Date: Pre-Calculus

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1 6.1: Verifying Trigonometric Identities Date: Pre-Calculus Using Fundamental Identities to Verify Other Identities: To verify an identity, we show that side of the identity can be simplified so that it is identical to the other side. In general, start with the more side of the equation and use the fundamental identities to transform this expression into the less complicated side of the equation. In addition, another technique is to rewrite the more complicated side in terms of and and then simplify this expression. Ex 1: Verify the identity: cscxtanx = secx Ex : Verify the identity: cosxcotx + sinx = cscx Some identities are verified by factoring to simplify a trigonometric expression. Ex 3: Verify the identity: sinx sinxcos x = sin 3 x

2 Another simplifying technique is to separate a single-term quotient into two terms using the property: a+b c = a c + b c Ex 4: Verify the identity: 1+cosθ sinθ = cscθ + cotθ If sums or differences of fractions with trigonometric functions appear on one side, it may be helpful to use the (LCD) to combine the fractions into a single fraction. Then other previously mentioned techniques will be used to finish simplifying. Note: This method is especially useful when the other side of the identity contains only one term. Ex 5: Verify the identity: sinx + 1+cosx = cscx 1+cosx sinx Homework: pg. 614 # 30(e)

3 6.1: Verifying Trigonometric Identities (Continued) Date: Pre-Calculus Some identities are verified using a technique that may remind you of rationalizing the denominator (Multiplying the conjugate). If there is a term like 1 + sinx in the denominator, multiply the numerator and denominator by (the conjugate). Ex 1: Verify the identity: cosx = 1 sinx 1+sinx cosx Remember, when all else fails, change everything into and and simplify. Ex : Verify the identity: secx+csc( x) secxcscx = sinx cosx

4 Some identities are not easily verified by working with only one side. In these cases, you can work with each side separately and show that both sides are equal to the same trigonometric expression. Ex 3: Verify the identity: = + 1+sinθ 1 sinθ tan θ Guidelines for Verifying Trigonometric Identities: o Work with each side of the equation of the other side. Start with the more complicated side and transform it in a step-by-step fashion until it looks exactly like the other side. o Analyze the identity and look for opportunities to apply the fundamental identities. o Try using one or more of the following techniques: Rewrite the more complicated side in terms of and Factor out the greatest common factor (GCF) Separate a single-term quotient into two terms Combine fractional expressions using the least common denominator (LCD) Multiply the numerator and denominator by a binomial factor that appears on the other side of the identity o Don t be afraid to stop and start over again if you are not getting anywhere. It is a puzzle! Ex 4: Verify the identity: 1+cosx 1 cosx = (cscx + cotx)

5 Ex 5: Verify the identity: (cosθ sinθ) + (cosθ + sinθ) = Ex 6: Verify the identity: (cos x sin x) 1 tan x = cos x Ex 7: Verify the identity: tanθ+cotθ cscθ = secθ

6 Ex 8: Rewrite cosx + tanx in terms of cosx 1+sinx Ex 9: Use the graph to make a conjecture about what the right side of the identity should be. Then prove your conjecture. cosx + cotxsinx cotx = Ex 10: Use the graph to make a conjecture about what the right side of the identity should be. Then prove your conjecture = secx+tanx secx tanx Homework: pg. 614 # pg. 614 #3 66(e), 67, 68

7 6.: Sum and Difference Formulas Date: Pre-Calculus Sum and Difference Formulas for Cosines and Sines: cos(α + β) = cosαcosβ sinαsinβ cos(α β) = cosαcosβ + sinαsinβ sin(α + β) = sinαcosβ + cosαsinβ sin(α β) = sinαcosβ cosαsinβ Ex 1: We know that cos30 =. Obtain this exact value using the identity cos30 = cos (90 60 ) and the difference formula for cosines. Ex : Find the exact value of cos70 cos40 + sin70 sin40 Ex 3: Verify the identity: cos(α β) cosαcosβ = 1 + tanαtanβ

8 Ex 4: Find the exact value of sin 5π 5π using the fact that = π + π Ex 5: Suppose that sinα = 4 for a quadrant II angle α and sinβ = 1 for a quadrant I angle β. Find the exact value of 5 each of the following: a) cosα c) cos (α + β) d) sin (α + β) b) cosβ Sum and Difference Formulas for Tangents tan(α + β) = tanα+tanβ 1 tanαtanβ tan(α β) = tanα tanβ 1+tanαtanβ Ex 6: Verify the identity: tan(x + π) = tanx Homework: pg. 63 # 38(e)

9 6.: Sum and Difference Formulas Continued Date: Pre-Calculus Ex 1: Derive the identity for tan (α + β) using by dividing numerator and denominator by cosαcosβ. tan(α + β) = sin(α+β) cos(α+β) Ex : Show that: tan (θ + π 4 ) = cosθ+sinθ cosθ sinθ

10 Ex 3: Verify the identity: sin(α+β) = (tanα+tanβ) sin(α β) tanα tanβ Ex 4: Given that cosα = 8 and α is in Quadrant IV and sinβ = 1 and β is in Quadrant III, find 17 a) cos(α + β) b) sin(α + β) c) tan(α + β) Homework: pg. 64 #40 68(e)

11 6.3: Half Angle Identity Examples (Continued) Date: Pre-Calculus Ex 1: Use cos10 to find cos105 Ex : If tanα = 4 and 180 < α < 70, find: 3 a) sin α b) cos α c) tan α

12 Ex 3: Verify that tanθ = sinθ 1+cosθ Ex 4: Write sin 4 x without powers of trigonometric functions greater than 1. Hint: Use the Power Reducing Identities from the lab: sin θ = 1 cosθ, cos θ = 1+cosθ Homework: pg. 635 #36 54, 56, 58, 59 6, 70 78(e)

13 6.4: Product-to-Sum and Sum-to-Product Formulas Date: Pre-Calculus Product-to-Sum Formulas: (Use these to write products of sines and/or cosines as sum or differences) sinαsinβ = 1 [cos(α β) cos(α + β)] cosαcosβ = 1 [cos(α β) + cos(α + β)] sinαcosβ = 1 [sin(α + β) + sin(α β)] cosαsinβ = 1 [sin(α + β) sin(α β)] *These formulas will be provided on tests and quizzes Ex 1: Verify the second formula cosαcosβ = 1 [cos(α β) + cos(α + β)] Ex : Express each of the following products as a sum or difference. a) sin5xsinx b) cos7xcosx Sum-to-Product Formulas: (Use these to write the sum or difference of sines and/or cosines as products) sinα + sinβ = sin α+β a β cos sinα sinβ = sin α β cosα + cosβ = cos α+β cosα cosβ = sin α+β cos α+β cos α β sin α β *These formulas will be provided on tests and quizzes

14 Ex 3: Verify the third formula α + cosβ = cos α+β. (Start with the right side and use the product-sumformulas). cos α β Ex 4: Express each sum as a product: a) sin7x + sin3x b) cos3x + cosx Ex 5: Verify the identity: cos3x cosx sin3x+sinx = tanx

15 Ex 6: Find the exact value of the expression sin ( π ) sin (5π ) 1 1 Ex 7: Verify the identity: cos3x cos5x sin3x+sin5x = tanx Homework: pg. 643 #6 1(e), 18 38(e), 40 44(e) part (a) only, 66

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17 6.5: Trigonometric Equations Date: Pre-Calculus To solve an equation containing a single trigonometric function: o Isolate the function on one side of the equation o Solve for the variable To find all solutions to a trigonometric function: o Solve to find all solutions on [0,π) for sine and cosine and [0, π) for tangent o Using the period for the function, write the general answers. (Use +πn for sine and cosine and +πn for tangent where n is an integer to allow for positive and negative angles). Ex 1: Solve the equation: 5sinx = 3sinx + 3 Equations Involving Multiple Angles: Ex : Solve the equation: tanx = 3 where 0 x < π

18 Ex 3: Solve the equation: sin x 3 = 1 where 0 x < π Trigonometric Equations Quadratic in Form Set Equal to Zero and Factor Ex 4: Solve the equation: sin x 3sinx + 1 = 0 where 0 x < π Ex 5: Solve the equation: 4 cos x 3 = 0 where 0 x < π

19 Ex 6: Solve the equation: sinxtanx = sinx where 0 x < π Using Identities to Solve a Trigonometric Equation: Ex 7: Solve the equation: sin x 3cosx = 0 where 0 x < π Ex 8: Solve the equation: cosx + sinx = 0 where 0 x < π Homework: pg. 656 # 74(e)

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21 6.5: Trigonometric Equations (Continued) Date: Pre-Calculus Ex 1: Solve the equation: sinxcosx = 1 where 0 x < π Ex : Solve the equation: cosx sinx = 1 where 0 x < π Using a Calculator to Solve Trigonometric Equation: Ex 3: Solve each equation, correct to four decimal places for 0 x < π a) tanx = b) sinx = 0.315

22 Ex 4: Solve the equation, correct to four decimal places, for 0 x < π: cos x + 5cosx + 3 = 0 Ex 5: Solve the equation sinxcosx + cosxsinx = for 0 x < π Ex 6: Solve the equation sin (x + π ) + sin (x π ) = 1 for 0 x < π 4 4 Homework: MathXL: Review