University of North Georgia Department of Mathematics

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1 Instructor: Berhanu Kidane Course: Precalculus Math 13 University of North Georgia Department of Mathematics Text Books: For this course we use free online resources: See the folder Educational Resources in Shared class files 1) (Book1) 2) Trigonometry by Michael Corral (Book 2) Other online resources: E Book: Tutorials: o / o o o Animation Lessons: o For more free supportive educational resources consult the syllabus 1

2 Trigonometric Identities (Book 2 page 65) Objectives: By the end of this section student should be able to Identify Fundamental or Basic Identities Find trigonometric values using the trig Identities Evaluate trigonometric functions Identities Equations: Three types 1) Conditional equations: These types of equations have finitely number of solutions. Example: a) 2xx 5 = 7xx, b) 3xx 2 4xx 6 = 0 2) Contradictions: These are equations that do not have solutions Examples: 2xx 1 = 2(xx 1) + 6 3) Identities: These types of equations hold true for any value of the variable Examples: (xx + 5)(xx 5) = xx² 25 Trigonometric Identities are identities of the Trigonometric equations. We use an identity to give an expression a more convenient form. In calculus and all its applications, the trigonometric identities are of central importance. Fundamental or Basic Trigonometric Identities Reciprocal Identities, Quotient Identities and Pythagorean Identities 1) Reciprocal identities (Page: 65) ssssss θθ =, aaaaaa cccccc θθ = cccccc θθ = tttttt θθ = ccccccθθ ssssssss cccccccc, and ssssss θθ =, and ccoooo θθ = ssssssss cccccccc tttttttt Proof: Follows directly from the definition of trig functions. 2) Quotient Identities tttttt θθ = ssssssss cccccccc cccccccc, and cccccc θθ = ssssssss Proof: Follows directly from the definition of trig functions. 2

3 3) Pythagorean Identities ( page: 66 & 67) a) ssssss²θθ + cccccc²θθ = b) + tttttt²θθ = ssssss²θθ c) + cccccc²θθ = cccccc ²θθ Proof: a) Let PP(xx, yy) be on the terminal side of the angle θθ. Then rr = xx 22 + yy 22 which implies that rr 22 = xx 22 + yy 22, ssssss θθ = yy rr, and cccccc θθ = xx rr And so, ssssss²θθ + cccccc²θθ = xx22 yy22 + = xx22 + yy 22 rr22 rr 22 rr 22 = rr22 = rr22 Example: Book 2: Example 3.1, 3.2, 3.3, 3.4, 3.5, 3.6 and 3.7 reading (67 69) Note: From a) it follows that: ssssss²θθ = cccccc²θθ and cccccc²θθ = ssssss²θθ, b) and c) are similarly proved. ssssss²θθ, "sine squared theta", means(ssssss θθ)² Example 1: a) Express sin θθ in terms of cos θθ b) Express cos θθ in terms of sin θθ c) Express tan θθ in terms of cos θθ, where θ in Quadrant II d) If tan θθ = 3 and θ is in Quadrant III, find sin θθ and cos θθ 2 e) If cos θθ = 1 and θ is in Quadrant IV, find all other trig values of θ 2 f) Use the basic trigonometric identities to determine the other five values of the trigonometric functions given that sin αα = 7/8 and cos αα > 0. g) xx is in quadrant II and ssssss xx = 1/5. Find cccccc xx and tttttt xx. Example 2: Prove the Pythagorean Identities b) and c) Example 3: Homework Reading page Examples 3.1, 3.2, 3.3, 3.4, 3.5, 3.6 and 3.7 (Book 2) Homework Exercises 3.1 page 70: # 1 21 odd numbers Examples YouTube Videos Trigonometric Identities 1) 2) 3) 3

4 4) Pythagorean Identity 5) Trigonometric Identities 6) Verifying more difficult Trig. Identities 4

5 7) Review Trig identities (1) Understanding Trig I 5

6 Further on Trigonometric Identities a) Co-Function Identities More on Trigonometry Identities ssssss ππ θθ = cccccc θθ, cccccc 22 tttttt ππ θθ = cccccc θθ, cccccc 22 ssssss ππ θθ = cccccc θθ, cccccc 22 ππ 22 ππ 22 ππ 22 θθ = ssssss θθ, θθ = ssssss θθ, θθ = ttttnn θθ Proof: By definition, referring to the figure above ssssss ππ 22 θθ = xx rr = cccccc θθ cccccc ππ θθ = yy = ssssss θθ 22 rr Similarly all the remaining cofunction identities follow 6

7 b) Even-Odd Identities ssssss( θθ) = ssssss θθ, cccccc( θθ) = cccccc θθ, tttttt( θθ) = tttttt θθ, cccccc( θθ) = cccccc θθ, ssssss( θθ) = ssssss θθ, cccccc( θθ) = cccccc θθ Proof: Let θθ be an angle of the 1 st Quadrant, then θθ is angle of 4 th Quadrant, see figure. rr = OOOO = OOOO. Now, ssssss( θθ) = yy rr = yy = ssssss θθ rr The rest of the Even Odd Identities for an angle of the 1 st Quadrant can be justified the same way. 4) Sum and difference formulas (page 71-73) a) ssssss (αα + ββ) = ssssss αα cccccc ββ + cccccccc ssssss ββ b) ssssss (αα ββ) = ssssss αα cccccc ββ cccccccc ssssss ββ c) cccccc ( αα + ββ) = cccccc αα cccccc ββ ssssss αα ssssss ββ d) cccccc (αα ββ) = cccccc αα cccccc ββ + ssssss αα ssssss ββ e) tttttt (αα + ββ) = f) tttttt (αα ββ) = tan αα +tan ββ 1 tan αα tan ββ tan αα tan ββ 1+tan αα tan ββ Example 1: Find the exact values of: a) ssssss 00 b) cccccc Example 2: Simplify the following expression. ssssss ( ππ 22 xx) tttttt(ππ 22 xx) ssssss(ππ 22 xx) Example: Book 2: Example 3.9, 3.10, 3., 3.12 reading (74 75) (Book 2) Homework Exercises 3.2 page 76 77: # 1 16 odd numbers 7

8 5) Double-angle formulas (page 78) a) ssssss (2222) = (θθ)cccccc(θθ) b) cccccc (2222) = cccccc 22 θθ ssssss 22 θθ c) cccccc (2222) = θθ d) cccccc (2222) = 22ssssss 22 θθ e) tttttt (2222) = tttttt 22 θθ Proof: Proof follows from Sum Difference Formulas Example: Book 2: Example 3.13, 3.14 Example 2: Given an angle for which ssssss(αα) = 33/55 in Quadrant III, determine the values for ssssss(22αα), cccccc (22αα), tttttt(22αα), ssssss(αα/22), cccccc(αα/22), and tttttt(αα/22). 6) Half-angle formulas (page 79 80) + cccccc θθ a) cccccc (θθ/22) = ± 22 cccccc θθ b) ssssss (θθ/22) = ± 22 c) tan θθ θθ = ± 1 cos 2 1+cos θθ Example: Book 2: Example 3.15 page 81 P roof: Proof follows from Double-angle Formulas Example 3: Find the Exact value of ssssss ππ 88. 7) Products as sums a) ssssss αα cccccc ββ = ½[ssssss (αα + ββ) + ssssss (αα ββ)] b) cccccc αα ssssss ββ = ½[ssssss (αα + ββ) ssssss (αα ββ)] c) cccccc αα cccccc ββ = ½[cccccc (αα + ββ) + cccccc (αα ββ)] d) ssssss αα ssssss ββ = ½[cccccc (αα + ββ) cccccc (αα ββ)] 8) Sums as products a) ssssss AA + ssssss BB = 22 ssssss ½ (AA + BB) cccccc ½ (AA BB) b) ssssss AA ssssss BB = 22 ssssss ½ (AA BB) cccccc ½ (AA + BB) c) cccccc AA + cccccc BB = 22 cccccc ½ (AA + BB) cccccc ½ (AA BB) d) cccccc AA cccccc BB = 22 ssssss ½ (AA + BB) ssssss ½ (AA BB) (Book 2) Homework Exercises 3.3 page 81: # 1 18 odd numbers 8

9 Example 4: Show that ssssss 22 xx + cccccc 22 xx = ssssss 22 xx cccccc 22 xx Example 5: Prove that a) tttttt yy ssssss yy = ssssss yy b) ssssss yy + ssssss yy cccccc 22 yy = cccccc yy c) tttttt xx + cccccc xx = ssssss xx cccccc xx d) +cccccc xx ssssss xx = ssssss xx cccccc xx e) tttttt(ππ xx) = tttttt (xx) f) tttttt 33 ππ + xx = cccccc (xx) 22 Example 6: Use the basic trigonometric identities to determine the other five values of the trigonometric functions given that: a) ssssss αα = 77/88 and cccccc αα > 00. b) xx is an angle in quadrant III and ssssss xx = / 33. c) xx is an angle in quadrant IV and tan x = -5. d) xx is in quadrant II and ssssss xx = /55. e) xx is in quadrant I and cccccc xx = /55. Example 7: Write AA ssssss bbbb + BB cccccc bbbb = aa ssssss(bbbb + cc) Solved Examples: 1) Simplify the following trigonometric expression. cccccc (xx) ssssss (ππ/22 xx) Use the identity ssssss (ππ/22 xx) = cccccc(xx) and simplify cccccc (xx) ssssss (ππ /22 xx) = cccccc (xx) cccccc (xx) = cccccc (xx) 2) Simplify the following trigonometric expression. [ssssss 4 xx cccccc 4 xx] / [ssssss 2 xx cccccc 2 xx] Factor the denominator [ssssss 4 xx cccccc 4 xx] / [ssssss 2 xx cccccc 2 xx] = [ssssss 2 xx cccccc 2 xx][ssssss 2 xx + cccccc 2 xx] / [ssssss 2 xx cccccc 2 xx] = [ssssss 2 xx + cccccc 2 xx] = 1 9

10 3) Simplify the following trigonometric expression. [ssssss(xx) ssssss 2 xx] / [1 + ssssss(xx)] Substitute sec (x) that is in the numerator by / cccccc (xx) and simplify. [ssssss(xx) ssssss 2 xx] / [1 + ssssss(xx)] = ssssss 2 xx / [ cccccc xx (1 + ssssss (xx) ] = ssssss 2 xx / [ cccccc xx + 1 ] Substitute ssssss 22 xx bbbb cccccc 22 xx, factor and simplify. = [ 1 cccccc 2 xx ] / [ cccccc xx + 1 ] = [ (1 cccccc xx)(1 + cccccc xx) ] / [ cccccc xx + 1 ] = 1 cccccc xx 4) Simplify the following trigonometric expression. ssssss ( xx) cccccc (ππ/22 xx) Use the identities ssssss ( xx) = ssssss (xx) and cccccc (ππ / 22 xx) = ssssss (xx) and simplify ssssss ( xx) cccccc (ππ/ 22 xx) = ssssss (xx) ssssss (xx) = ssssss 22 xx 5) Simplify the following trigonometric expression. ssssss 22 xx cccccc 22 xx ssssss 22 xx Factor ssssss 22 xx out, group and simplify ssssss 22 xx cccccc 22 xx ssssss 22 xx = ssssss 2 xx ( 1 cccccc 2 xx ) = ssssss 4 xx 6) Simplify the following trigonometric expression. tttttt 44 xx + 22 tttttt 22 xx + Note that the given trigonometric expression can be written as a square tttttt 44 xx + 22 tttttt 22 xx + = ( tttttt 22 xx + ) 22 We now use the identity 1 + tan 2 x = sec 2 x tttttt 44 xx + 22 tttttt 22 xx + = ( tttttt 22 xx + ) 22 = ( sec 2 x ) 2 = sec 4 x 7) Add and simplify. / [ + cccccc xx] + / [ cccccc xx] In order to add the fractional trigonometric expressions, we need to have a common denominator / [ + cccccc xx] + / [ cccccc xx] = [ 1 cccccc xx cccccc xx ] / [ [1 + cccccc xx] [1 cccccc xx] ] = 2 / [1 cccccc 2 xx] = 2 / ssssss 2 xx = 2 cccccc 2 xx 10

11 8) Write ( 4 4 ssssss 2 xx ) without square root for (ππ/ 22) < xx < ππ. Factor, and substitute ssssss 22 xx bbbb cccccc 22 xx ( 4 4 ssssss 2 xx ) = 4(1 ssssss 2 xx ) = 2 cccccc 2 xx = 2 cccccc (xx) Since (ππ/ 22) < xx < ππ, cccccc xx is less than zero and the given trigonometric expression simplifies to = 22 cccccc (xx) 9) Simplify the following expression. [ ssssss 44 xx] / [ + ssssss 22 xx] Factor the denominator, and simplify [ ssssss 44 xx] / [ + ssssss 22 xx] = [1 ssssss 2 xx] [1 + ssssss 2 xx] / [1 + ssssss 2 xx] = [ ssssss 22 xx] = cccccc 22 xx 10) Add and simplify. / [ + ssssss xx] + / [ ssssss xx] Solution : Use a common denominator to add [ + ssssss xx] + [ ssssss xx] = [ ssssss xx + + ssssss xx] [ ( + ssssss xx)( ssssss xx)] = 22 / [ ssssss 22 xx ] = 22 / cccccc 22 xx = 22 ssssss 22 xx ) Add and simplify. cccccc xx cccccc xx ssssss 22 xx factor cccccc xx out ; cccccc xx cccccc xx ssssss 22 xx = cccccc xx ssssss 22 xx = cccccc xx cccccc 22 xx = cccccc 33 xx

12 12) Simplify the following expression. tttttt 22 xx cccccc 22 xx + cccccc 22 xx ssssss 22 xx Use the trigonometric identities tttttt xx = ssssss xx / cccccc xx and cccccc xx = cccccc xx / ssssss xx to write the given expression as tttttt 22 xx cccccc 22 xx + cccccc 22 xx ssssss 22 xx = (ssssss xx / cccccc xx) 22 cccccc 22 xx + (cccccc xx / ssssss xx) 22 ssssss 22 xx and simplify to get: = ssssss 22 xx + cccccc 22 xx = 13) Simplify the following expression. ssssss ( ππ 22 xx) tttttt(ππ 22 xx) ssssss(ππ 22 xx) Use the identities ssssss ( ππ xx) = cccccc xx, 22 tttttt(ππ xx) = cccccc xx and 22 ssssss(ππ xx) = cccccc xx to write 22 the given expression as ssssss ππ xx tttttt 22 ππ xx ssssss 22 ππ xx = cccccc xx cccccc xx cccccc xx 22 = cccccc xx cccccc xx ssssss xx cccccc xx = cccccc xx cccccc 22 xx/ssssss xx = /ssssss xx cccccc 22 xx/ssssss xx = ( cccccc 22 xx) /ssssss xx = ssssss 22 xx/ssssss xx = ssssss xx 14) Show that cccccc 44 xx ssssss 44 xx = cccccc(2222) Practice Problems 12