5.8 Trigonometric Equations

Transcription

1 5.8 Tigonometic Equations To calculate the angle at which a cuved section of highwa should be banked, an enginee uses the equation tan =, whee is the angle of the bank and v is the speed limit on the cuve, in kilometes pe hou. The equation tan = is an eample of a tigonometic equation. It will be used in Eample 6. A tigonometic equation is an equation that contains one o moe tigonometic functions. Othe eamples of tigonometic equations include the following. sin 2 + cos 2 = 1 2cos 1 = 0 The equation sin 2 + cos 2 = 1 is an identit. Recall that a tigonometic identit is an equation that is tue fo all values of the vaiable fo which the epessions on both sides of the equation ae defined. The equation 2cos 1 = 0 is not an identit and is onl tue fo cetain values of. To solve a tigonometic equation that is not an identit means to find all values of the vaiable,, that make the equation tue. INVESTIGATE & INQUIRE 1. To gaph the sstem = and = 9, use the Y= edito to ente Y1 = 2X + 3 and Y2 = 9 X. Gaph the equations in the standad viewing window. 2. Use the intesect opeation to find the coodinates of the point of intesection. 3. Equate the ight sides of the two equations fom question 1 and solve the esulting equation fo algebaicall. 4. Compae the value of ou found in question 3 to the value of ou found in question 2. Eplain ou findings. 402 MHR Chapte 5

2 5. To gaph the equations = 4cos and = 1 + 2cos, use the Y= edito to ente Y1 = 4cos(X) and Y2 = 1 + 2cos(X). Using the mode settings, select the degee mode. Using the window vaiables, adjust the window to and 5 5 b setting Xmin = 0, Xma = 360, Ymin = 5, and Yma = 5. Displa the gaph. 6. Use the intesect opeation to find the coodinates of the points of intesection. 7. Equate the ight sides of the two equations fom question 5, and solve the esulting equation fo cos algebaicall. 8. If 0 360, what values of, in degees, give the value of cos fom question 7? 9. Compae the values of ou found in question 8 to the values of ou found in question 6. Eplain ou findings. EXAMPLE 1 Solving a Tigonometic Equation Solve 2 sin θ 1 = 0 on the inteval 0 θ 360. SOLUTION 1 Pape-and-Pencil Method 2 sin θ 1 = 0 Add 1 to both sides: 2 sin θ = 1 Divide both sides b 2 : sin θ = 1 2 The sine of an angle is positive in the fist and second quadants. 0 θ θ 0 sin θ = The solutions ae 45 and 135. In adians, the solutions ae π 4 and 3 4π. Use the memo aid CAST to emembe which tigonometic atios ae positive in each quadant. 5.8 Tigonometic Equations MHR 403

3 SOLUTION 2 Gaphing-Calculato Method Gaph the elated tigonometic function = 2 sin 1 fo Using the mode settings, select the degee mode. Find the -intecepts using the zeo opeation. The gaph intesects the -ais at 45 and 135. The solutions ae 45 and 135. In adians, the solutions ae θ = π 4 and 3. 4π The window vaiables include Xmin = 0, Xma = 360, Ymin = 3, Yma = 1. EXAMPLE 2 Solving a Tigonometic Equation Find the eact solutions fo 3cos θ = cos θ + 1, if 0 θ 360. SOLUTION 1 Pape-and-Pencil Method 3cos θ = cos θ + 1 Subtact cos θ fom both sides: 2cos θ = 1 Divide both sides b 2: cos θ = 1 2 The cosine of an angle is positive in the fist and fouth quadants. 0 θ θ 0 cos θ = The eact solutions ae 60 and 300. In adians, the eact solutions ae π 3 and 5 3π. 404 MHR Chapte 5

4 SOLUTION 2 Gaphing-Calculato Method Gaph = 3cos and = cos + 1 in the same viewing window fo Use the intesect opeation to detemine the coodinates of the points of intesection. The gaphs intesect at = 60 and = 300. The window vaiables include Xmin = 0, Xma = 360, Ymin = 4, Yma = 4. The solutions ae 60 and 300. In adians, the solutions ae π 3 and 5 3π. Note that an altenative gaphing-calculato method fo solving Eample 2 involves ewiting 3cos = cos + 1 as 2cos 1 = 0. The equation 2cos 1 = 0 can be solved b gaphing = 2cos 1 and using the zeo opeation to find the -intecepts. The window vaiables include Xmin = 0, Xma = 360, Ymin = 4, Yma = 2. EXAMPLE 3 Solving b Factoing Solve the equation 2cos 2 cos 1 = 0 on the inteval 0 2π. SOLUTION 2cos 2 cos 1 = 0 Facto the left side: (2cos + 1)(cos 1) = 0 Use the zeo poduct popet: 2cos + 1 = 0 o cos 1 = 0 2cos = 1 cos = 1 The solutions ae 0, 2 π, 4 π, and 2π. 3 3 cos = 1 2 = 2 π o 4 π 3 3 = 0 o 2π The solution can be modelled gaphicall. The window vaiables include Xmin = 0, Xma = 360, Ymin = 3, Yma = Tigonometic Equations MHR 405

5 EXAMPLE 4 Solving b Factoing Solve the equation 2sin 2 7sin + 3 = 0 fo 0 2π. Epess answes as eact solutions and as appoimate solutions, to the neaest hundedth of a adian. SOLUTION 2sin 2 7sin + 3 = 0 Facto the left side: (sin 3)(2sin 1) = 0 Use the zeo poduct popet: sin 3 = 0 o 2sin 1 = 0 sin = 3 2sin = 1 sin = 1 2 Thee is no solution to sin = 3, since all values of sin ae 1 o 1. The eact solutions ae π 6 and 5 6π. = π 6 o 5 6π The appoimate solutions ae 0.52 and EXAMPLE 5 Using a Tigonometic Identit Solve 6cos 2 sin 5 = 0 fo Round appoimate solutions to the neaest tenth of a degee. SOLUTION Use the Pthagoean identit cos 2 = 1 sin 2 to wite an equivalent equation that involves onl the sine function. 6cos 2 sin 5 = 0 Substitute 1 sin 2 fo cos 2 : 6(1 sin 2 ) sin 5 = 0 Epand: 6 6sin 2 sin 5 = 0 Simplif: 6sin 2 sin + 1 = 0 Multipl both sides b 1: 6sin 2 + sin 1 = 0 Facto the left side: (2sin + 1)(3sin 1) = 0 Use the zeo poduct popet: 2sin + 1 = 0 o 3sin 1 = 0 2sin = 1 o 3sin = 1 sin = 1 2 o sin = 1 3 = 210 o 330 = 19.5 o The solutions ae 19.5, 160.5, 210, and MHR Chapte 5

6 EXAMPLE 6 Bank Angles Enginees use the equation tan = to calculate the angle at which a cuved section of highwa should be banked. In the equation, is the angle of the bank and v is the speed limit on the cuve, in kilometes pe hou. a) Calculate the angle of the bank, to the neaest tenth of a degee, if the speed limit is 100 km/h. b) The fou tuns at the Indianapolis Moto Speedwa ae banked at an angle of 9.2. What is the maimum speed though these tuns, to the neaest kilomete pe hou? SOLUTION a) tan = = = 2.6 The angle of the bank is 2.6, to the neaest tenth of a degee. b) = tan = tan v = 224 0tan 00 v = 224 0tan v = 190 The maimum speed is 190 km/h, to the neaest kilomete pe hou. Ke Concepts To solve a tigonometic equation that is not an identit, find all values of the vaiable that make the equation tue. Tigonometic equations can be solved a) with pape and pencil using the methods used to solve algebaic equations b) gaphicall using a gaphing calculato Answes can be epessed in degees o adians. 5.8 Tigonometic Equations MHR 407

7 Communicate You Undestanding 1. Descibe how ou would solve 2sin 3 = 0, Descibe how ou would solve 2cos 2 3cos + 1 = 0, 0 2π. Justif ou method. 3. Eplain wh the equation cos 2 = 0 has no solutions. Pactise A 1. Solve each equation fo 0 2π. a) sin = 0 b) 2cos + 1 = 0 c) tan = 1 d) 2 sin + 1 = 0 e) 2cos 3 = 0 f) 2sin + 3 = 0 2. Solve each equation fo a) sin + 1 = 0 b) 2 cos 1 = 0 c) 2sin 3 = 0 d) 2 cos + 1 = 0 e) 2sin + 1 = 0 f) tan = 1 3. Solve each equation fo a) 2cos 2 7cos + 3 = 0 b) 3sin = 2cos 2 c) 2sin 2 3sin 2 = 0 d) sin 2 1 = cos 2 e) tan 2 1 = 0 f) 2sin 2 + 3sin + 1 = 0 g) 2cos 2 + 3sin 3 = 0 4. Solve each equation fo 0 2π. Epess answes as eact solutions and as appoimate solutions, to the neaest hundedth of a adian. a) sin 2 2sin 3 = 0 b) 2cos 2 = sin + 1 c) 2sin cos + sin = 0 d) sin 2 = 6sin 9 e) sin 2 + sin = 0 f) cos = 2sin cos 5. Solve fo on the inteval Round appoimate solutions to the neaest tenth of a degee. a) 4cos 3 = 0 b) 1 + sin = 4sin c) 6cos 2 cos 1 = 0 d) 9sin 2 6sin + 1 = 0 e) 16cos 2 4cos + 1 = 0 f) 6cos 2 + sin 4 = 0 6. Solve fo on the inteval 0 2π. Give the eact solution, whee possible. Othewise, ound to the neaest hundedth of a adian. a) sin 3sin cos = 0 b) 6cos 2 + cos 1 = 0 c) 8cos cos = 3 d) 8sin 2 10cos 11 = MHR Chapte 5

8 Appl, Solve, Communicate 7. Refaction of light The diagam shows how a light a bends as it tavels fom ai into wate. The bending of the a is a pocess known as efaction. In the diagam, i is the angle of incidence and is the angle of efaction. Fo wate, the inde of efaction, n, is defined as follows. n = a) The inde of efaction of wate has a value of Fo a a with an angle of incidence of 40, find the angle of efaction, to the neaest tenth of a degee. b) State an estictions on the value of in the above equation. Eplain. B sin i sin 8. Isosceles tiangle a) Show that the aea, A, of an isosceles tiangle with base b and equal angles, can be found using the equation A = b 2 tan. 4 i b) Fo an isosceles tiangle with A = 40 cm 2 and b = 8 cm, use the equation to find, to the neaest tenth of a degee. 9. Application In ight tiangle ABC, BD AC, AC = 4, and BD = 1. a) Show that sin A cos A = 1 4. b) Descibe how ou would use a gaphing calculato to find the measue of A, in degees. c) Find the eact measue of A, in degees. A B D C 10. Solve fo on the inteval 0 2π. Give the eact solution, if possible. Othewise, ound to the neaest hundedth of a adian. a) tan 2 4tan = 0 b) tan 2 5tan + 6 = 0 c) tan 2 4tan + 4 = 0 d) 6tan 2 7tan + 2 = Solve fo on the inteval Check ou solutions. a) 2sin tan tan 2sin + 1 = 0 b) cos tan 1 + tan cos = a) Gaph = sin and = 0.5 fo b) Fo what values of is sin 0.5? c) Fo what values of is sin 0.5? 5.8 Tigonometic Equations MHR 409

9 13. a) Communication Wite a shot paagaph, including eamples, to distinguish the tems tigonometic identit and tigonometic equation. b) Inqui/Poblem Solving If (cos + sin ) 2 + (cos sin ) 2 = k, fo what value(s) of k is the equation an identit? C 14. Solve each equation on the inteval a) sin 2 = 1 b) cos 2 = 1 c) 2sin 2 = 1 d) 2cos 2 = 1 e) 2 cos 2 = 1 f) 2sin 2 = 3 g) 2cos 2 = 3 h) sin 2 = 0 i) 2sin 0.5 = Technolog Use a gaphing calculato to find the eact solutions, in degees, fo a) cos 2 = cos b) sin 2 = 2cos c) tan = sin Technolog a) Descibe how ou would use a gaphing calculato to solve the equation sin (cos ) = 0, b) Find the eact solutions to this equation. A CHIEVEMENT Check Knowledge/Undestanding Thinking/Inqui/Poblem Solving Communication Application A stom dain has a coss section in the shape of an isosceles tapezoid. The shote base and each of the two equal sides measue 2 m, and is the angle fomed b the longe base and each of the equal sides. 2 m a) Wite an epession fo the aea, A, of the coss section in tems of sin and cos. b) Descibe how ou would use a gaphing calculato to find the value of, in degees, if the aea of the coss section is 5 m 2. c) Find the value of, to the neaest degee. 2 m 2 m NUMBER Powe Place the digits fom 1 to 9 in the boes to make the statements tue. Use the ode of opeations. = 2 ( + ) = 2 + = MHR Chapte 5