Trigonometric Functions. Concept Category 3

Transcription

1 Trigonometric Functions Concept Category 3

2 Goals 6 basic trig functions (geometry) Special triangles Inverse trig functions (to find the angles) Unit Circle: Trig identities a b c

3 The Six Basic Trig functions a adjacent Cos c hypotenuse b opposite Sin c hypotenuse b opposite Tan a adjacent Sec Csc Cot 1 cos 1 sin 1 tan Sin b Tan Cos a C is always opposite of the right angle

4 The sides of a right -angled triangle are given special names: The hypotenuse, the opposite and the adjacent. The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90 o.

5 I will start posting extra practice for CC1, CC, and CC3 online if you want to start preparing for the finals Suggestion: Focus on passing two CCs first

6 Sin Cos Opp Hyp Adj Hyp hypotenuse opposite Tan Opp Adj adjacent

7 Trig Functions For example evaluate sin 40 using sin key You should get:

8 Some Sine Practice Function Try each of these on your calculator: sin 55 = sin 10 = sin 87 = 0.999

9 Where to use these trig functions (ratios).

10 Goal Problem: cm x cm How do we solve x???

11 34 15 cm x cm How do we solve x??? Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse What trig ratio uses Opposite and Hypotenuse? SINE Set up the equation and solve: (15) (15) sin 34 x 15 (15)Sin 34 = x 8.39 cm = x

12 Ex) 53 1 cm x cm Ask yourself: In relation to the angle, what pieces do I have? Ask yourself: Opposite and adjacent What trig ratio uses Opposite and adjacent? (1) Tan 53 = x (1) 1 (1)tan 53 = x 15.9 cm = x tangent Set up the equation and solve:

13 x cm Ask yourself: In relation to the angle, what pieces do I have? cm Adjacent and hypotenuse Ask yourself: What trig ratio uses adjacent and hypotnuse? cosine Set up the equation and solve: (x) Cos 68 = 18 (x) x (x)cos 68 = 18 cos 68 cos 68 X = 18 cos 68 X = cm

14 Ex) From a point 80m from the base of a tower, the angle of elevation is 8. How tall is the tower? x 80 8 Using the 8 angle as a reference, we know opp. and adj. sides. Use opp adj tan tan 8 = x (tan 8 ) = x 80 (.5317) = x x m

15 Ex ) A ladder that is 0 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75, how far will Coach Jarvis have to crawl to get to the front door when he falls off the ladder (assuming he falls to the base of the ladder)? 0 75 building Using the 75 angle as a reference, we know hypotenuse and adjacent side. adj Use cos cos 75 = hyp x 0 x 0 (cos 75 ) = x 0 (.588) = x x ft.

16 Ex 3. When the sun is 6 above the horizon, a building casts a shadow 18m long. How tall is the building? x 6 18 shadow Using the 6 angle as a reference, we know opposite and adjacent side. opp Use x tan tan 6 = adj (tan 6 ) = x 18 (1.8807) = x x m

17 Inverse Trig Function to find the Angle Inverse Sine Function Using sin -1 (inverse sine): If = sin θ then sin -1 (0.7315) = θ angle

18 More Examples: 1. sin x = find angle x. x = sin -1 (0.1115) sin = x = 6.4 o. cos x = find angle x x = cos -1 (0.8988) cos = x = 6 o

19 cm 4 cm θ This time, you re looking for angle ɵ Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse What trig ratio uses opposite and hypotenuse? sine Set up the equation : Sin θ = /4 Use the inverse function to find an angle Sin -1 (/4) = θ = θ

20 Example C cm Find an angle that has a tangent (ratio) of /3 B 3cm A Process: I want to find an ANGLE I was given the sides (ratio) Tangent is opp/adj TAN -1 (/3) = 34 Angle A

21 1. H 14 cm We have been given the adjacent and hypotenuse so we use COSINE: Cos A = 6 cm A C Cos A = Cos C = h a 14 6 Cos C = C = cos -1 (0.486) C = 64.6 o adjacent hypotenuse

22 . Find angle x x 3 cm A 8 cm O Tan A = Tan x = Tan x =.6667 o a 8 3 Given adj and opp need to use tan: Tan A = opposite adjacent x = tan -1 (.6667) x = 69.4 o

23 Do it Now: C Solve the right triangle: c =? angle B =? angle A =? B 3 c A

24 C Solution: 3 (hypotenuse) = (leg) + (leg) c = 3 + c = c = 13 c = 13 c 3.6 B Pythagorean Theorem c A

25 continued Then use a calculator to find the measure of B: tan o Then find A: ma = mb 56.3

26 Goal Problem: Space Shuttle: During its approach to Earth, the space shuttle s glide angle changes. When the shuttle s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.

27 Solution: You know opposite and adjacent sides. Which trig ratio (function) can you use? Glide = x tan x = opp. distance to runway adj. 59 miles Use correct ratio altitude 15.7 miles tan x = Substitute values Use inverse function: Tan-1 (15.7/59) 14.9 When the space shuttle s altitude is about 15.7 miles, the glide angle is about 14.9.

28 Part b) When the space shuttle is 5 miles from the runway, its glide angle is about 19. Find the shuttle s altitude at this point in its descent. Round your answer to the nearest tenth. The shuttle s altitude is about 1.7 miles. Glide = 19 tan 19 = opp. tan 19 = h distance to runway adj. 5 5 tan 19 = h 5 5 miles altitude h Use correct ratio Substitute values 5 Isolate h by multiplying by h Approximate using calculator

29 Types of Angles The angle that your line of sight makes with a line drawn horizontally. Angle of Elevation Line of Sight Angle of Elev ation Horizontal Line Angle of Depression Horizontal Line Angle of Depression Line of Sight

30 Nov8 Warm-up: Do These Now Find all key features; sketch : f( x) f( x) x x x 1 x3 x x

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33 Unit Circle Introduction Reminder: Pythagorean Theorem Angle

34 On an x-y plane Thus,a b c x y r

35 It s about a circle and a triangle. r = radius The chosen angle is always attached to the origin (0,0)

36 Trig Functions + xy coordinate plane You need to remember these formulas for the final

37 Since x y r if you think about it : x y r... Conics r r r x y 1... radius 1 r r From yesterday : Thus (cos ) (sin ) 1

38 Proof! o o (cos 35 ) (sin 35 )? o o (cos 5 ) (sin 5 )? o o (cos 300 ) (sin 300 )?

39 How about these guys? x y r x x x x y r y y y Rewrite the equations using Trig functions

40 Proof 1 (tan 30 0 ) (sec30 0 )??? But, your calculator doesn t have a sec key.

41 1 1 (tan 30 ) ( ) cos30 0 0

42 Unit Circle (calculator practice) Try : sin 0 o cos 0 o sin 45 o cos 45 o sin 360 cos 360 o o

43 Unit Circle (calculator practice ) Try : o sin 0. o cos 0. o sin o cos o sin o cos 0.707

44 Special Triangles: Find the 6 Trig Functions (Ratios) for each

45 sin 30 a 1 a o 3a 3 cos 30? calculator a o a 1 tan 30? calculator 3a 3 sin 60 cos 60 tan 60 sin 45 cos 45 tan 45 o o o o o o o calculator o sin 30? o cos 30? o tan 30?

46 Not just for fractions

47 Unit Circle: circle with center at (0, 0) and radius = 1 x y r r 1 (-1,0) (0,1) (0,-1) (1,0) o sin o cos o tan y r x r y x So points on this circle must satisfy this equation.

48 This about this. cos sin o o x x r 1 y y r 1 x y Thus : ( x, y) (cos,sin ) o o

49 Handout :

50 Angle first 150 π 90 / 10 π / 3 3π / π / 6 1 0,1 1,0 180 π 0 0 1, π / π / 4 7π / π / 3 5π 300 / 3 3π 70 / -1 0, 1 π 60 / 3 π / 4 45 π / π / 6 330

51 sin 0 o sin 30 o cos 0 o cos 30 o tan 0 o tan 30 o (1,0) 30 o

52 sin 45 o sin 60 o cos 45 o cos 60 o tan 45 o tan 60 o 45 o

53 sin 90 cos 90 tan 90 o o o (0,1)

54 r 1 and ( x, y) cos,sin 3 1, -1 1,0 3 1,, π, , , , / 0, , , 1 1, or, 330 1/ 0 3 1, 1, 1,0 3 1,

55 ( x, y) ( xy, ) ( x, y) ( x, y)

56 (cos30 o,sin 30 o ) in 4 quadrants

57 (cos45 o,sin 45 o ) in 4 quadrants

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59 Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian Section is the 4.1, measure Figure of 4.5, a central Illustration angle of that intercepts arc s Arc equal Length, in length pg. to 49 the radius r of the circle. In general, s r 59

60 Section 4.1, Figure 4.7, Common Radian Angles, pg. 49 Radian Measure 60

61 Formulas: Conversions Between Degrees and Radians To convert degrees to radians, multiply degrees by. To convert radians to degrees, multiply radians by

62 Use Use Change 140º to Radians 180 Change 7 3 to degrees degree to rads (radians) rads (radians) to degrees

63 Before. 3 1, -1 1,0, , , , 45 30, 0 3 1, 1 1,0 3 1, 10, , , , 3 1,,

64 Angle : Degree v. s. Radian 3 1, -1 1,0 3 1,, π, 5π / 6 7π / 6 1 3, 1 3, 1 π / π / 3 3π / 4 5π / 4 4π / 3-1 0, / 0, , π / 3 π / , π / 6 11π / 6 7π / 4 5π / 3 3π /, 1/ 0 3 1, 1, 1,0 3 1,

65 3 1, -1 1,0, π 5π / 6 1 3, 1 π / π / 3 3π / 4 0,1 1 3, π / 3 π / 4 π / 6, 0 3 1, 1 1,0 3 1,, 7π / 6 5π / 4 4π / 3 1 3, -1 7π / 4 5π / 3 3π / 0, 1 11π / 6 1 3,, 3 1,

66 : : : Reference Angles Can you see a pattern???? Quadrant I II III IV ,, ,, ,, 3 3 3

67 Example Find the exact value of the following: cos 3 4 Reference Angle: Cosine of Reference Angle: cos 4 4 o Quadrant of Reference Angle: Second Quadrant Sign of Cosine in Second Quadrant:,,,, Therefore: 3 cos 4

68 Do Now:

69 Summary of CC3 6 basic trig functions: sin, cos, tan, etc Special Triangles (exact values) How to solve for an angle; values (ratio) given a trig function Unit Circle: x y r and r 1 Unit Circle + Special Triangles + Trig Functions Radian converting to Degree (of an angle)

70 CC1 Final Suggestions: Do finding limits given a graph and using limits to sketch a graph first: they are the least time consuming; then sketch rational graph and finding limits given a piecewise function; then IROC or AROC

71 CC1 : More Final Review Problems a) Sketch a graph given the description: lim f ( x) 0 lim f ( x) 0 lim f ( x) x x x lim f ( x) f (0) 0 lim f ( x) x x4 lim f( x) x4 b) Find key features and sketch: y x 7x6 4x 7x c) Find the slope and the equation of the tangent line (AROC) for : 14 f ( x) when x 4 x 3

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73 ( x 3) f (4) f ( x) f ( a) x 3 x 3 x3 ( x3) AROC x a x 4 x 4 x 4 14 x 6 x 8 ( x3) ( x3) x 8 1 ( x 4) 1 x 4 x 4 ( x 3) x 4 ( x 3) x 4 x 3 Slope lim x4 x 3 7 Equation : y mx b y x b (4) b 7 7

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75 Practice Now No Calculator

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79 CC3 - Section 3 Verifying Identities: 1. Solving equation? PEMDAS reverse. Factor an expression, add fractions, square a binomial, or create s monomial denominator, if possible. 3. Use the fundamental identities, whenever possible. 4. Convert all terms to sines and cosines.

80 csc *RECIPROCAL IDENTITIES* 1 1 sec 1 cot sin cos tan sin sin cosec tan *QUOTIENT IDENTITIES* sin cos cot cos sin *PYTHAGOREAN IDENTITIES* tan 1 sec cos 1 1cot csc EVEN-ODD IDENTITIES sin cos cos tan tan cosec sec sec cot cot

81 Where did our pythagorean identities come from?? Do you remember the Unit Circle? What is the equation for the unit circle? x + y = 1 What does x =? What does y =? (in terms of trig functions) cos θ + sin θ = 1 Pythagorean Identity!

82 Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos θ sin θ + cos θ = 1. cos θ cos θ cos θ tan θ + 1 = sec θ Quotient Identity another Pythagorean Identity Reciprocal Identity

83 Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin θ sin θ + cos θ = 1. sin θ sin θ sin θ 1 + cot θ = csc θ Quotient Identity a third Pythagorean Identity Reciprocal Identity

84 More on Pythagorean Identities sin + cos = 1 sin = 1 - cos cos = 1 - sin tan + 1 = sec tan = sec - 1 cot + 1 = csc cot = csc - 1

85 Guidelines for Verifying/Simplifying Identities: 1. Start with the most complicated side of the equation.. Factor an expression, add fractions, square a binomial, or create a monomial denominator, if possible. 3. Use the fundamental identities, whenever possible. 4. Convert all terms to sines and cosines. 85

86 Example : Verify the identity sec 1 sin. sec sec 1 (tan 1) 1 Start with the more complicated side. Pythagorean Identity sec sec tan sec tan (cos ) Simplify. Reciprocal Identity sin (cos ) cos sin Quotient Identity Simplify. 86

87 Graphing Utility: Verify the identity sec 1 sin. sec Table: Identical Values Identical Graphs Graph:

88 Example: Verify the identity (tan 1)(cos 1) tan. (tan 1)(cos 1) (sec )( sin ) Pythagorea n Identity sin cos Reciprocal Identity sin cos tan Rule of Exponents Quotient Identity 88

89 Happy Thursday!!! Practice: One page per pair Pick at least problems from each section. Use your notes from yesterday Your Semester final will include CC1 and from Semester1

90 Verify sec 1 sin sec sec 1 sec tan sec tan cos 1) Choose the left side since it is more complex ) Trig Identity 3) Reciprocal Function 4) Change to sine and cosine sin cos cos 1 sin

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92 Verify 1) Choose the right side since it is more complex 1 1 sec 1sin x 1sin 1 1sin x 1 1 sin x 1sin x1 sin x 1sin x1 sin x ) Make one fraction 1 sin x1sin 3) Simplify 4) Trig identity 5) Can t leave in denominator! sec x x x (1 sin)(1 sin x) 1 sin x cos x

93 More Practice: With Complex Fractions 1)Verify: 1 1 tan cos x x Solution: x 1 tan x 1 cos x 1 cos x sin cos x cos x cos x cos x )Verify: tan tan cot x cot x x x 1 cos x Solution: sin cos sin cos tan x cot x cos x sin x sin x cos x tan x cot x sin cos sin cos cos x sin x sin xcos x sin x cos x 1 cos x cos x 1 cos sin x cos x 1 x x x x x x x x x

94 Example) Verify the statement: sin csc cos sin Substitute using reciprocal identity sin csc cos sin sin We are done! We've shown the LHS equals the RHS 1 cos sin 1 cos sin Using the Pythagorean Identities: sin + cos = 1 thus sin = 1 - cos

95 Happy Wed.!!!! 1/5 th CC3 Mastery Check Practice Verify the Identities: D]

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101 In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match.

102 Happy Thursday CC3 Quick Check next Tuesday (no notes) Practice Quick Check today (notes ok): 1 minutes for problems 1 and only Version A v.s. Version B Critique: 10 minutes - Turn to the person behind you (with same version) It is ok to have different solution pathways! Then do problem 3 on the practice quiz and turn it in to me x (csc xcot x)

103 Critique Check list: Convert to same units (all sin; all cos; sin &cos) Fraction add, subt Fraction mult, div (complex fraction) Trig identities Factoring Common Factors (reverse distribution) Difference of Square Trinomial ax bx c Rationalization

104 Double-Angle and Half-Angle Identities

105 Double-Angle Formulas Formula for sine: sin x sin xcos x Formula for cosine: cosx cos x sin x 1sin x cos x 1 Formula for tangent: tan x tan x 1 tan x

106 Using Double Angle Formla Verify the identity. sin x cos 4x 1 sin x cos x 1 Substitute sin x 1 sin x 1 1 sin x 11

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109 Now you try: Verify the identity

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111 Monday 1/30 th : Use the Double Angle Formulas (and all the basic formulas) to Verify the Identities & Solve the Equations (solve for missing angles) Tomorrow + Wed: Sum and Difference Angles Formulas Thursday: Quick Check (on gradebook)

112 Sum and Difference Formulas

113 tan tan Formulas sin u v sin ucos v cos usin v sin u v sin u cos v cos usin v u u cos u v cos u cos v sin usin v cos u v cos u cos v sin usin v v tan uv v tan u tan v 1 tan utan v tan u tan v 1 tan utan v sin( u v) cos( u v)

114 Double angle formula: let A = B Cos ( A + B ) = Cos A Cos B - Sin A Sin B

115 Double angle formula A = B Sin ( A + B) = Sin A Cos B + Cos A Sin B

116 Double angle formula A = B tan ( A + B ) = tan A + tan B 1- tan A tan B

117 Try these now

118 Solutions

119 Practice: Finish yesterday s practice problems (right side solving equations) Or try this DOK3 problem:

120 Solution

121 Evaluating Trigonometric Expressions Solve the equation for 0 < x <. sin x cos x sin x 1 sin x sin x sin x 1 0 sin x 1 sin x 1 0 sin x 1 sin x 1 1 sin x 5 3 x,, 6 6

122 Evaluating Trigonometric Expressions Solve the equation for 0 < x <. cos x cos x 0 cos x 1 cosx 0 cos x cos x 3 0 cos x 3 cos x 1 0 cos x 3 cos x 1 cos x 3 x

123 More Example: Verify the identity. sin x cos x 1 sin x sin sin x x sin xcos x cos x cos x sin xcos x 1 sin xcos x 1 sin x

124 Example: Triple-Angle Formula Write cos 3x in terms of cos x. Solution: cos 3x = cos(x + x) = cos x cos x sin x sin x Addition formula = ( cos x 1) cos x ( sin x cos x) sin x Double-Angle Formulas = cos 3 x cos x sin x cos x Expand

125 cont. Solution = cos 3 x cos x cos x (1 cos x) = cos 3 x cos x cos x + cos 3 x cont d Pythagorean identity Expand = 4 cos 3 x 3 cos x Simplify

126 /7/17 Cosine & Sine Graphs

127 CC3 Graphs of Sine and Cosine Functions sketch the graphs of basic sine and cosine functions. use amplitude and period to help sketch the graphs of sine and cosine functions. sketch translations of graphs of sine and cosine functions. use sine and cosine functions to model real-life data.

128 3 1, ( 1, 0), 3, , 3 7 6, , y (0, 1) (0, 1) 1, 3 0 0, , 3, x (1, 0) 3, 1,

129 Evaluate the Sine Curve using the unit circle Section 4.5, Figure 4.4, Graph of Sine Curve, pg. 87 Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 17

130 Sine ~ Table Chart: Input x (angles) Output f(x) (values) 0 Sin(0) = 0 45 Sin(45) = Sin(90) =1 135 Sin(135) = (which is 0) 0

131 Sine ~ Parent Graph

132 Sine Function: Parent T-chart (simplified)

133 Sine Function Parent Graph

134 Vertical Transformation: Algebraically This method works for all graphs: exponential, log, quadratics, rational, etc.

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137 Horizontal Transformation Algebraically

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141 Sketch : a) y sin( x) 1 b) y 3sin( x ) 4

142 a) Vertically transformed Final Graph

143 b) Vertically transformed Final

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145 a) y 3cos( x) b) y 3cos( x) c) y 3cos( x ) 3

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147 sin Recall that tan. cos Tangent Function Since cos θ is in the denominator, when cos θ = 0, tan θ is undefined. This π intervals, offset by π/: { π/, π/, 3π/, 5π/, } Let s create an x/y table from θ = π/ to θ = π/ (one π interval), with 5 input angle values. θ sin θ cos θ tan θ θ tan θ π/ 1 0 und π/ und π/4 1 π/ π/4 1 π/4 1 π/ 1 0 und π/ und

148 θ Graph of Tangent Function: Periodic tan θ tan θ Vertical asymptotes where cos θ = 0 tan sin cos π/ Und (- ) π/4 1 3π/ π/ 0 π/ 3π/ θ 0 0 π/4 1 π/ Und( ) One period: π tan θ: Domain (angle measures): θ π/ + πn Range (ratio of sides): all real numbers (, ) tan θ is an odd function; it is symmetric wrt the origin. Domain, tan( θ) = tan(θ)

149 Tangent Function Parent Graph

150 Graph of Cotangent Function: Periodic cot θ Vertical asymptotes where sin θ = 0 cos cot sin θ cot θ 0 π/4 1 π/ 0 3π/ -π π/ π/ π 3π/ 3π/4 1 π cot θ: Domain (angle measures): θ πn Range (ratio of sides): all real numbers (, ) cot θ is an odd function; it is symmetric wrt the origin. Domain, tan( θ) = tan(θ)

151 Graph of the Secant Function 1 The graph y = sec x, use the identity sec x. cos x At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y y sec x Properties of y = sec x 1. domain : all real x x k ( k ). range: (, 1] [1, +) 3. period: 4. vertical asymptotes: x k k y cos x x

152 Summary of Graph Characteristics sin θ csc θ cos θ sec θ tan θ cot θ opp hyp 1.sinθ adj hyp Def n 1. sinθ sinθ cosθ cosθ.sinθ о Period Domain Range Even/Odd y r π (, ) r.y π θ πn x r π (, ) r y π θ π +πn y x π θ π +πn x y π θ πn 1 x 1 or [ 1, 1] csc θ 1 or (, 1] U [1, ) All Reals or (, ) sec θ 1 or (, 1] U [1, ) All Reals or (, ) All Reals or (, ) odd odd even even odd odd

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154 Sin(x) Think 1 Period as 1 Cycle Cos(x)

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166 Given the cosine graph below, can you create an equation?

167 What about this one? What is the equation?

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170 Application (DOK3)

171 If we decide to use cosine as the parent graph, then this is one period/cycle: from to 8 which is 6 seconds. If using sine as the parent then the values change.

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178 Review of CC3 Verify Identities: Solve the equations:

179 Verify the identities: Solve the equations: 3 cos x 1 sin x cos x cos x sin x DOK3] Verify the identities:

180 Verify the identities: Solve the equations: DOK3:

181 Mock Quiz: 0 minutes Verify the Identities: Solving the Equations: 3 cos x 1 sin x cos x cos x sin x Create an Equation (Model):

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195 Half-Angle Identities sin a 1 cos a cos a 1 cos a

196 Sum and Difference Formulas: sin( u v) sin ucos v cos usin v Signs are the same sin( u v) sin ucos v cos usin v Example: Verify the identity x sin cos x. sin x sin cos x cos sin x (1) cos x(0)sin x cos x 196

197 Sum and Difference Formulas (continued): cos( u v) cos ucos v sin usin v Signs are opposite cos( u v) cos ucos v sin usin v Signs are the same tan( u v) tan u tan v 1 tan utan v Signs are the same tan( u v) tan u tan v 1 tan utan v Opposite signs 197

198 Double-Angle Formulas: sin u sin u cos u cos cos sin cos u 1 u u u 1sin u Example: Verify the identity sin10 sin 5 cos5. sin10 sin( 5 ) sin 5 cos5 198

199 Double-Angle Formulas (continued): tan u tan u 1 tan u Example: 3 Verify the identity tan3 3tan tan. 13tan tan3 tan( ) tan tan 1 tan tan tan tan 1 tan tan ( tan ) 1 1 tan tan 3 tan tan 1tan tan Sum Formula Double-Angle Formula 3 3tan tan 13tan 199

200 Half-Angle Formulas: Example: sin u 1 cos u cos u 1 cos u tan u 1 cos u sin u sin u 1 cos u Use the half-angle formula to find the exact value of sin.5. sin.5 sin 45 1 cos

201 3 1, -1 1,0 3 1,, π, 5π / 6 7π / 6 1 3, 1 3, 1 π / π / 3 3π / 4 5π / 4 4π / 3-1 0, / 0, , π / 3 π / , 1 1, or, π / 6 11π / 6 7π / 4 5π / 3 3π / 1/ 0 3 1, 1, 1,0 3 1,

202 Most Common Application: r x y x y r r cos sin r θ y y tan 1 x x

203 Do you remember 30º, 60º, 90º triangles? Now they are really! Important

204 Do you remember 30º, 60º, 90º triangles? Now they are really! Important Even more important Let a = 1

205 Do you remember 30º, 60º, 90º Let a = 1 triangles? Cos30 Sin30 1 3

206 Do you remember 30º, 60º, 90º triangles? Cos60 Sin

207 Do you remember 45º, 45º, 90º triangles? When the hypotenuse is 1 The legs are Cos45 Sin45 1

208 The Unit Circle A circle with radius of 1 Equation x + y = 1 0,1 cos, sin 1,0 1,0 0, 1

209 The Unit Circle with Radian Measures

210 Some common radian measurements These are the Degree expressed in Radians

211 The Unit Circle: Radian Measures and Coordinates

212 Why does the book use t for an angle? Since Radian measurement are lengths of an arc of the unit circle, it is written as if the angle was on a number line. Where the distance is t from zero. Later when we graph Trig functions it just works better.

213 Lets find the Trig functions if Think where this angle is on the unit circle Tan Sin Cos 3 Cos Sin Tan 3, 1

214 Find the Trig functions of Think where this angle is on the unit circle Tan Sin Cos Cot Csc Sec

215 How about 4 4,, Tan Sin Cos

216 There are times when Tan or Cot does not exist. At what angles would this happen?

217 There are times when Tan or Cot does not exist., 3

218 If think of the domain of the trig functions, there are some limits. Look at the unit circle. If x goes with Cos, then what are the possible of Cos? It is the same with Sin?

219 Which of the three applications is to be used depends on what measurements of a triangle are known: ??

220 Definition of a Periodic Function A function f is periodic if there exist a positive real number c such that f(t + c) = f(t) for all values of t. The smallest c is called the period.

221 Even Function ( Trig. ) Cos (- t) = Cos (t) and Sec( -t) = Sec (t) Also Sin(-t) = -sin (t) and Csc (-t) = - Csc (t) Tan(-t) = -Tan (t) and Cot(-t) = - Cot (t)

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223 In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match.

224 Establish the following identity: cosec cot Let's sub in here using reciprocal identity and quotient identity We worked on LHS and then RHS but never moved things across the = sign combine fractions Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom sin cosec cot 1 cos 1 cos sin sin sin 1 cos sin 1 cos FOIL denominator 1 cos sin 1 cos sin 1 cos 1 cos 1 cos sin 1 cos sin 1 cos 1 cos sin 1 cos sin sin 1 cos 1 cos sin sin

225 Hints for Establishing Identities Get common denominators If you have squared functions look for Pythagorean Identities Work on the more complex side first If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities Have fun with these---it's like a puzzle, can you use identities and algebra to get them to match! MathXTC

226 4 4 4 o o 3 ans : cot 30 3; sec Acknowledgement a 1 1 x 4 4 x x x 3 3 ] lim b] lim x0 x o o I wish to thank Shawna Haider from Salt Lake Community x c] lim College, dutah ] lim x5 x 4 3 x3 USA for her hard work in creating this PowerPoint. 45,135 5 x 6 3 x 3 x 0 o 1 o x 10, 40 o Shawna has kindly given permission for this resource g] f ( xto ) be downloaded x 4 x from and for it to be modified to suit the Western h] f ( x) x 5 x 3 Australian Mathematics Curriculum. a] lim f ( x) x8 b] f ( 8) e f x x x ] ( ) f ] f ( x) x x 1 1 3) Stephen Corcoran Head of Mathematics St Stephen s School Carramar c] lim f ( x) x d] f ( ) )( x ) x 3 78

227 Using the identities you now know, find the trig value. 1.) If cosθ = 3/4, find secθ.) If cosθ = 3/5, find cscθ. sec 1 cos sin cos 1 sin sin sin 16 5 sin 4 5 csc 1 sin

228 3.) sinθ = -1/3, find tanθ tan 1 sec tan 1 (3) tan 8 tan 4.) secθ = -7/5, find sinθ tan 8

229 Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. a) cos sin tan cos sin sin cos cos sin cos cos sin cos 1 cos b) cot 1 sin cos sin cos 1 cos sin 1 cos 1 sin sec csc 5.4.5

230 Simplifing Trigonometric Expressions c) (1 + tan x) - sin x sec x (1 tanx) sinx 1 cosx 1 tanx tan x sinx cosx 1 tan x tanx tanx sec x d) csc x tan x cot x 1 sinx sinx cos x cosx sinx 1 sinx sin x cos x sinxcos x 1 sinx 1 sinx cos x 1 sinx cos x sinx cos x 1

231 Simplify each expression. 1 sin cos sin 1 sin sin cos 1 cos sec 1 cos x sin x sin x cos x 1 cos x cos x sin x sin x cos x sin x sin x sin x cos x sin x sin x 1 sin x csc x

232 Simplifying trig Identity Example1: simplify tanxcosx sin tanx x cosx cos x tanxcosx = sin x

233 Simplifying trig Identity Example: simplify sec x csc x 1 cos sec x 1 csc 1 x cos x sin x sin x = cos x = x = tan x sinx 1

234 Simplifying trig Identity Example: simplify cos x - sin x cos x cos cos x - x 1 sin - sin x x cos x = sec x

235 Example Simplify: = cot x (csc x - 1) Factor out cot x = cot x (cot x) Use pythagorean identi = cot 3 x Simplify

236 Simplify: Example = sin x (sin x) + cos x cos x = sin x + (cos x) cos x cos x cos x = sin x + cos x cos x = 1 cos x = sec x Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean iden Use reciprocal identity

237 Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

238 1 Practice

239 One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let s see an example of this: tan x sin x cos x substitute using each identity csc x 1 sin x simplify sin x 1 cos x sin x 1 cos x Simplify: tan xcsc x sec x 1 cos x 1 1 cos x sec x 1 cos x

240 Another way to use identities is to write one function in terms of another function. Let s see an example of this: Write the following expression in terms of only one trig function: cos xsin x1 = 1sin xsin x1 = sin xsin x This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute. sin xcos x 1 cos x 1sin x

241 (E) Examples Prove tan(x) cos(x) = sin(x) LS tan x cos x LS sin cos x x cos x LS sin x LS RS 41

242 (E) Examples Prove tan (x) = sin (x) cos - (x) RS RS RS RS RS RS RS sin x sin x cos x sin x cos x sin sin cos tan LS x cos x x x 1 x 1 cos x 4

243 (E) Examples Prove tan x 1 1 tan x sin x cos x LS LS LS LS LS LS LS 1 tan x tan x sin x 1 cos x sin x cos x sin x cos x cos x sin x sin x sin x cos x cos x cos x sin x sin x cos x cos x sin x 1 cos x sin x RS 43

244 (E) Examples Prove sin x 1 cos x 1 cos x LS LS LS LS sin x 1 cos x x 1 cos 1 cos x ( 1 cos x)( 1 cos x) ( 1 cos x) 1 cos x LS RS 44