To derive the other Pythagorean Identities, divide the entire equation by + = θ = sin. sinθ cosθ tanθ = 1

Transcription

1 Syllabus Objetives: 3.3 The student will simplify trigonometri expressions and prove trigonometri identities (fundamental identities). 3.4 The student will solve trigonometri equations with and without tehnology. Identity: a statement that is true for all values for whih both sides are defined 3 x+ 8 = 3x+ 3 Example from algebra: Reiproal Identities sin = os = tan = s se ot s = se = ot = sin os tan Quotient Identities sin os tan = ot = os sin Reall: Unit Cirle r =, x = os, y = sin ( x, y) Note: sin = ( sin ) Pythagorean Theorem: x + y = Pythagorean Identity: si n + os = To derive the other Pythagorean Identities, divide the entire equation by sin and then by os : sin os + = sin sin sin sin os + = os os os Pythagorean Identities sin + os = + ot = s tan + = se Simplifying Trigonometri Expressions: Look for identities Change everything to sine and osine and redue Ex: Use basi identities to simplify the expressions. os a) ot ( os ) ot = sin + os = sin = os sin b) tan ( s ) sin tan = s = os sin Page of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

2 Ex: Simplify the expression ( s x+ )( s x ) Use algebra: os + ot = s ot = s x. Solving Trigonometri Equations Isolate the trigonometri funtion. Solve for x using inverse trig funtions. Note There may be more than one solution or no solution. Ex3: Solve the equation 4sin x 4 = 0 in the interval [ 0,π ). Find values of x for whih x sin and x sin = = : x= Exploration: Consider a right triangle. a Φ b Note that φ and are omplementary. Write the trig funtions for eah angle. What do you notie? sinφ = os = osφ = sin = tanφ = ot = sφ = se = seφ = s = otφ = tan = Page of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

3 **The trig funtions of φ are equal to the ofuntions of, when Cofuntion Identities: π sin 90 = os or sin = os π os 90 = sin or os = sin π se 90 = s or se = s π s 90 = se or s = se π tan 90 = ot or ot = tan π ot 90 = tan or ot = tan φ and are omplementary. Exploration: Consider an angle,, and its opposite, as shown in the oordinate grid. Compare the trig funtions of eah angle. z z x y sin =, sin ( ) = os =, os ( ) = tan, tan ( ) s =, s ( ) = se =, se ( ) = ot, ot ( ) **Cosine and seant are the only EVEN ( f ( x) f ( x) ) ( f ( x) = f ( x) ). Odd-Even Identities: ( ) ( ) ( ) ( ) ( ) ( ) sin = sin and s = s tan = tan and ot = ot os = os and se = se = = = = = trig funtions. All the rest are ODD Page 3 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

4 Ex4: Simplify the expression sin( x) s( x). sin( x) = sin x s( x) = s x Simplifying Trigonometri Expressions: Simplify using the following strategies. Note that the equations in bold are the trig identities used when simplifying. All of the other steps are algebra steps. Ex5: Simplify the expression by fatoring. os x + os x( sin x) sin x+ os x = Ex6: Simplify the expression by ombining frations. sin x+ os x = s x = sin x sin x os x osx sinx Solving Trigonometri Equations: Solve using the following strategies. Find all solutions for eah equation in the interval [ 0,π ). Ex7: Solve the equation by isolating the trig funtion. osx = 0 These are values of x where the osine is equal to. Page 4 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

5 Ex8: Solve the equation by extrating square roots. 4sin x 3= 0 These are values of x where the sine is equal to 3 ±. Ex9: Solve the equation by fatoring. Set equal to zero. os x + osx = Fator. Set eah fator equal to zero. Solve eah equation. Note: It may be easier to use u-substitution with u = os x to help students visualize the equation as a quadrati equation that an be fatored. Ex0: Solve the equation by fatoring. se xsin x se x = 0 Fator out GCF. Use zero produt property. Solve eah equation. Note: It is possible for an equation to have no solution. Ex: Solve by rewriting in a single trig funtion. Substitute Pyth. Identity. sin x+ 3osx = 3 sin os sin os + = = Simplify algebraially. Fator and solve. Page 5 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

6 Rewrite. Ex: Solve using trig substitutions. 3 sin x tan x os x = Rewrite sin x tan x os x =. Ex3: Find the approximate solution using the alulator. 4osx = Isolate the trig funtion. os x = 4 To find x, we need to find the inverse osine of ¼. x = os 4 When solving an equation in the interval [ 0,π ), be sure to be in Radian mode. x.38 You Try: Make the suggested trigonometri substitution and then use the Pythagorean Identities to write the resulting funtion as a multiple of a basi trig funtion. 4 x, x = os Refletion: Explain the relationship between trig funtions and their ofuntions. Syllabus Objetive: 3.3 The student will simplify trigonometri expressions and prove trigonometri identities. Trigonometri Identity: an equation involving trigonometri funtions that is a true equation for all values of x Tips for Proving Trigonometri Identities:. Manipulate only one side of the equation. Start with the more ompliated side.. Look for any identities (use all that you have learned so far). 3. Change everything to sine or osine. Page 6 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

7 4. Use algebra (ommon denominators, fatoring, et) to simplify. 5. Eah step should have one hange only. 6. The final step should have the same expression on both sides of the equation. Note: Your goal when proving a trig identity is to make both sides look idential! For all of the following examples, prove that the identity is true. The trig identities used in the substitutions are in bold. 3 os x = sin x os x Ex: Start with the right side (more ompliated). sin x+ os x = os x = sin x Ex: + = se sinx + sinx Start with the left side. x Combine frations. Simplify. Trig substitution. Identity. sin x+ os x = os x = sin se x os x = x Ex3: Start with the left side. Trig substitution. tan x + os x = tan x Trig substitution. sin x+ os x = os x = sin x Trig substitution. se x os x = Multiply. Page 7 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

8 Identitiy. os x Ex4: se x+ tan x = sinx Start with the left side. sin x tan x os x = Change to sine/osine. Combine frations. Multiply num/den by onjugate. Multiply. Trig substitution. Simplify. sin x+ os x = os x = sin x se Ex5: sin = se Start with left side. Split the fration. Simplify. Trig substitution. Identity. os se = sin + os = sin = os Challenge: Try to prove the identity above in another way. You Try: Prove the identity. ( x x) ( x x) os sin + os sin = Refletion: List at least 5 strategies you an use when proving trigonometri identities. Page 8 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

9 Syllabus Objetive: 3.3 The student will simplify trigonometri expressions and prove trigonometri identities (sum and differene identities). Reall: and = 00 = = = 4 So in general, a+ b a + b ( + 3) = ( 5) = 5 So in general, a+ b a + b = 3 Sum and Differene Identities tan sin u± v = sin uos v± osusin v os u± v = osuosv sin uosv tanu± tan v ± = tanutanv ( u v) Note: Be areful with +/ signs! Simplifying Expressions with Sum and Differenes. Rewrite the expression using a sum/differene identity.. Simplify the expression and evaluate if neessary. Ex: Write the expression as the sine of an angle. Then give the exat value. π π π π sin os os sin 4 4 sin u v = sin uosv osusinv Evaluating Trigonometri Expressions with Non-Speial Angles. Rewrite the angle as a sum or differene of two speial angles.. Rewrite the expression using a sum/differene identity. 3. Evaluate the expression. or Ex: Find the exat value of o s = os95 = os u+ v = osuosv sin usinv. 3. os50 os45 sin50 sin 45 = Page 9 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

10 Ex3: Write as one trig funtion and find an exat value.. tan( u v). 3. tan u+ tanv + = tanutanv tan80 + tan55 tan80 tan55 Evaluating Trig Funtions Given Other Trig Funtion(s) Ex4: Find os( u v) 5 3π 4 π given osu =, π < u < and sin v =, 0 < v<. 7 5 os u v = osuosv+ sin usinv We must find o sv and sin u. Draw the appropriate right triangles in the oordinate plane. 5 3π 4 π osu =, π < u < : sin v =, 0 < v< : 7 5 y y 5 u x 7 v 5 4 x Use the Pythagorean Theorem to find the missing sides. In Quadrant III, sine is negative, so sin u =. In Quadrant I, osine is positive, so os u v = osuosv+ sin usinv = 3 osv =. 5 Proving Identities sin( α + β) Ex5: Verify the identity. = tanα + tan β osαos β Start with the left side. sin u+ v = sin uosv+ osusinv Trig substitution: Page 0 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

11 Split the fration: Simplify: Trig substitution: You Try: Verify the ofuntion identity sin π = os using the angle differene identity. Refletion: Give an example of a funtion for whih f ( a+ b) = f ( a) + f ( b) for all real numbers a and b. Then give an example of a funtion for whih f ( a+ b) f ( a) + f ( b) for all real numbers a and b. Syllabus Objetive: 3.3 The student will simplify trigonometri expressions and prove trigonometri identities (double angle and power-reduing identities). Ex: Derive the double angle identities using the sum identities.. sin( u) = sin ( u+u ). os( u) = os( u+u ) 3. tan( u) = tan( u+u ) Double Angle Identities sin = sin os os = os sin tan tan = tan There are two other ways to write the double angle identity for osine. Use the Pythagorean identity. Page of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

12 sin + os = sin = os os = sin os = os sin os = sin os = os Evaluating Double-Angle Trigonometri Funtions 3π Ex: Find the exat value of o su given ot u = 5, < u< π. y u 5 x u will be in Quadrant IV and forms a right triangle as labeled. Using the Pythagorean Theorem, we have 5 Double Angle Identity: osu = os u sin u os u =, sin u = 6 6 Note: If u is in Quadrant IV, 3 π < u < π, then for u we have whih is in Quadrant I. So it makes sense that osu is positive. Solving Trigonometri Equations Ex3: Find the solutions to 4sin x os x = in [ ) 0,π. Rewrite the equation. Trig substitution. sin x = sin xos x Isolate trig funtion. Solve for the argument. Beause the argument is x, we must revisit the domain. [ 0,π ) is the restrition for x. So 0 x < π. Therefore,. Page of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

13 Solve for x. Rewriting a Multiple Angle Trig Funtion to a Single Angle Ex4: Express sin3x in terms of sin x. Rewrite argument as a sum Sum identity Double angle identities Pythagorean identity Simplify Verifying a Trig Identity tan Ex5: Verify sin =. + tan Start with left side. Pythagorean identity Rewrite in sines/osines Simplify Double angle identity Reall: os = os sin os = sin os = os Solving for sin and os, we an derive the power reduing identities. os = sin os sin = os = os + os os = Page 3 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

14 os sin os tan = = = os + os + os Power Reduing Identities os sin = + os os = os tan = + os Ex6: Express Rewrite as a produt Power reduing identity Multiply Power reduing identity 5 os x in terms of trig funtions with no power greater than. You Try:. Find the solutions to osx sinx 0. Verify os α otα tanα sin α =. + = in [ ) 0,π. Refletion: How do you onvert from a osine funtion to a sine funtion? Explain. Page 4 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

15 Syllabus Objetive: 3.3 The student will simplify trigonometri expressions and prove trigonometri identities (half angle identities). Reall: os u sin = Let =. We have u osu sin = u Solving for sin, we have u osu sin =±. All of the other half-angle identities an be derived in a similar manner. Half-Angle Identities u osu sin =± u + osu os =± u osu tan =± + os u Note: There are others for tangent. u osu tan = sin u u sin u tan = + os u Note: The ± will be deided based upon whih quadrant u lies in. Evaluating Trig Funtions π Ex: Find the exat value of os. Rewrite as a half angle Half angle identity Evaluate Choose sign π is in Quadrant I, where osine is positive. Solving a Trig Equation Ex: Solve the equation sin Half-angle identity Square both sides Pythagorean identity x = in [ ) sin x 0,π. Page 5 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

16 Set equal to zero Fator Zero produt property You Try:. Find the exat value of tan.5.. Solve the equation x sin os x 0 + = in [ ) 0,π. Refletion: Explain why two of the half-angle identities do not have +/ signs. Syllabus Objetive: 3.5 The student will solve appliation problems involving triangles (Law of Sines). Deriving the Law of Sines: Consider the two triangles. C C b a h b h a A B A B h h In the aute triangle, sin A = and sin B =. b a h In the obtuse triangle, sin( π B) = sin B =. a Solve for h. h = bsin A and h = asin B sin B sin A Substitute. asin B = bsin A an be rewritten as = b a sin The same type of argument an be used to show that C sin B sin A = =. b a Page 6 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

17 Law of Sines: The ratio of the sine of an angle to the length of its opposite side is the same for all three angles of any triangle. b C a sin A sin sin = B = C or a = b = a b sin A sin B sinc A B Solving a Triangle: finding all of the missing sides and angles Note: The Law of Sines an be used to solve triangles given AAS and ASA. Ex: Solve the triangle ΔABC given that B = 5, C = 5, and b= 9. 9 C a A B We are given AAS, so we will use the Law of Sines. sin B sinc = Solve for : b Find m A using the triangle sum. m A= sin B sin A = Solve for a: b a ΔABC: m A=, m B =, m C =, a =, b=, = Ambiguous Case: When given SSA, there ould be triangles, triangle, or no triangles that an be reated with the given information. Ex: Solve the triangle Given SSA, use Law of Sines. Solve for A. ΔABC (if possible) when m C = 54, a = 0, = 7. sinc sin A = a There is no possible triangle with the given information. Page 7 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

18 Ex: Solve the triangle Given SSA, use Law of Sines. Solve for B. ΔABC (if possible) when m C = 3, b= 46, = 9. sinc sin B = b Note that the alulator only gives the aute angle measure for B. There does exist an obtuse angle B with the same sine. m B = = 5. This is also an appropriate measure of an angle in a triangle, so there are triangles that an be formed with the given information. Triangle Triangle Triangle : m A=, m B =, m C =, a =, b=, = Triangle : m A=, m B =, m C =, a =, b=, = Appliation Problems. Draw a piture!. Use the Law of Sines to solve for what is asked in the problem. Ex3: The angle of elevation to a mountain is Approximate the height of the mountain. 5. After driving 3 miles, the angle of elevation is z h (not drawn to sale) First, find :. Therefore, the third angle in the small triangle = 5.5. Using the law of sines, we know that Now we an use right triangle trig to find h: Page 8 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

19 You Try: Solve the triangle ΔABC (if possible) when m B = 98, b= 0, = 3. Refletion: Explain why SSA is the ambiguous ase when solving triangles. Syllabus Objetive: 3.5 The student will solve appliation problems involving triangles (Law of Cosines). Law of Cosines: For any triangle, ABC C = a + b abosc b = a + aosb a = b + bosa A b a B Note: In a right triangle, os90 ( 0) = a + b ab = a + b ab = a + b (Pythagorean Theorem) The Law of Cosines an be used to solve triangles when given SAS or SSS. Ex: Solve the triangle ABC when m A= 49, b= 4, & = 5. Note: The given information is SAS. Use a = b + bosa. Now that we have a mathing pair of a side and angle, we an use the Law of Sines. a b = sin B = B sin A sin B or Page 9 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

20 Now find the two possibilities for m C using the triangle sum: or Sine is the shortest side, it must be opposite the smallest angle. So m C =. m A=, m B =, m C =, a =, b=, = Ex: Solve the triangle ABC when a = 3, b= 5, & = 8. Note: The given information is SSS. Use a = b + bosa. (Or any of them!) Now that we have a mathing pair of a side and angle, we an use the Law of Sines. a B sin A = or Now find the two possibilities for m C using the triangle sum: or Sine is the shortest side, it must be opposite the smallest angle. So m C =. m A=, m B =, m C = 6.7, a =, b=, = Appliation Problems. Draw a piture!. Use the Law of Cosines to solve for what is asked in the problem. Ex3: A plane takes off and travels 60 miles, then turns 5 and travels for 80 miles. How far is the plane from the airport? (not drawn to sale) Page 0 of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4

21 Using the piture, we an find the angle in the triangle to give us SAS. Use the Law of Cosines: = a + b abosc Area of a Triangle h Reall: A= bh sin = h = sin ; A= bsin b h Formula for the Area of a Triangle Given SAS: h b A= absinc Ex4: Find the area of triangle ABC shown. B A C We will use A= asin B. Heron s Area Formula Semi-Perimeter: a+ b+ s = Area of a Triangle Given SSS: A= s( s a)( s b)( s ) Ex5: Find the area of the triangle with side lengths 5 m, 6 m, and 9 m. a+ b+ Semiperimeter: s = s = A= s( s a)( s b)( s ) You Try: Two ships leave port with a 9 angle between their planned routes. If they are traveling at 3 mph and 3 mph, how far apart are they in 3 hours? Refletion: Can there be an ambiguous ase when using the Law of Cosines? Explain why or why not. Page of Prealulus Graphial, Numerial, Algebrai: Pearson Chapter 4