MQ Maths B Yr 11 Ch 06 - Jacaranda | ??2010-06-30238 Maths Quest Maths B Year 11 for Queensland ......

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Transcript of MQ Maths B Yr 11 Ch 06 - Jacaranda | ??2010-06-30238 Maths Quest Maths B Year 11 for Queensland ......

  • 6syllabussyllabusrrefefererenceenceTopic: Periodic functions and

    applications

    In thisIn this chachapterpter6A Simple trigonometric

    equations6B Equations using radians6C Further trigonometric

    equations6D Identities6E Using the Pythagorean

    identity

    Trigonometricequations

    MQ Maths B Yr 11 Ch 06 Page 237 Thursday, October 25, 2001 3:59 PM

  • 238 M a t h s Q u e s t M a t h s B Ye a r 1 1 f o r Q u e e n s l a n d

    IntroductionSudhira is a keen fisherman. The ideal depth for fishing in Sudhiras favourite tidal lakeis 3 metres. The depth of water in the lake can be found using the equation

    D = 5 4 sin t

    where t is the time in hours after midnight. What is the best time of day for Sudhira tofish?

    To solve this problem we need to solve a trigonometric equation

    Simple trigonometric equationsFrom your earlier work on trigonometry, you will be familiar with problems of the type: Find the size of the angle marked in the figure at right.

    The solution to this problem is set out as:

    cos =

    cos =

    = 46.

    The equation cos = is an example of a trigonometric equation. This trigonometric

    equation had to be solved in order to find the size of the angle in the triangle. In thisparticular case we knew that the angle was acute from the triangle that was drawn.

    In the earlier chapter on graphing periodic functions we saw that the cos functionwas periodic. This means that there are values of , other than the one already found

    for which cos = . There will, in fact, be an infinite number of solutions to this

    trigonometric equation, so for practical reasons we are usually given a domain withinwhich to solve the equation. This domain will often be in the form 0 360,meaning that we want solutions within the first positive revolution.

    If the trigonometric ratio is positive the calculator will give a first quadrant answer. To complete the solution we need to consider all quadrants for which the trigonometric ratio is positive.

    In the case of cos = the cosine ratio is positive in

    the first and fourth quadrants. We found earlier that the first quadrant solution to this equation was 46. The fourth quadrant solution will therefore be 360 46 = 314.

    For a negative trigonometric ratio we solve the corresponding positive equation to find a first quadrant angle to use, then find thecorresponding angles in the negative quadrants.

    p6---

    13 cm

    9 cm

    adjhyp---------

    913------

    913------

    913------

    y

    x

    Sinepositive

    Allpositive

    Tangentpositive

    Cosinepositive

    913------

    MQ Maths B Yr 11 Ch 06 Page 238 Thursday, October 25, 2001 3:59 PM

  • C h a p t e r 6 Tr i g o n o m e t r i c e q u a t i o n s 239

    In the earlier chapter we also found that we were able to find exact values of specialangles using the triangles on the next page.

    Solve the following trigonometric equations over the domain 0 360, correct to the nearest degree.a sin = 0.412

    b tan =

    THINK WRITE

    a Write the equation. a sin = 0.412Use your calculator to find the first quadrant angle.

    First quadrant angle = 24

    The sine ratio is positive in the first and second quadrants.Find the second quadrant angle by subtracting 24 from 180.

    180 24 = 156

    Write the answer. = 24 or 156

    b Write the equation. b tan =

    Use your calculator to find the first quadrant angle.

    First quadrant angle = 20

    The tangent ratio is negative in the second and fourth quadrants.Find the second quadrant angle by subtracting 20 from 180 and the fourth quadrant angle by subtracting 20 from 360.

    180 20 = 160360 20 = 340

    Write the answer. = 160 or 340

    411------

    12

    y

    x24

    180 24

    3

    4

    5

    14

    11------

    2

    y

    x20

    360 20

    180 20

    3

    4

    5

    1WORKEDExample

    MQ Maths B Yr 11 Ch 06 Page 239 Thursday, October 25, 2001 3:59 PM

  • 240 M a t h s Q u e s t M a t h s B Ye a r 1 1 f o r Q u e e n s l a n d

    These special angles should be used where possible in the solution to a trigonometricequation. They are used when we recognise any of the values produced by the triangles.

    sin 30 = cos 30 = tan 30 =

    sin 45 = cos 45 = tan 45 = 1

    sin 60 = cos 60 = tan 60 =

    Similarly, we must be aware of when theboundary angles should be used in the solutionof the equation. Remember from the work onthe unit circle that y = sin , x = cos and

    tan = .

    45

    45

    2

    1

    1

    30

    60

    2

    1

    Line of bisection

    12--- 3

    2------- 3

    3-------

    22

    ------- 22

    -------

    32

    ------- 12--- 3

    Solve the equation cos q = over the domain 0 360.

    THINK WRITE

    Write the equation. cos =

    Use the special triangles to find the first quadrant angle.

    First quadrant angle = 30

    The cosine ratio is negative in the second and third quadrants.Find the second quadrant angle by subtracting 30 from 180 and find the third quadrant angle by adding 30 to 180.

    180 30 = 150180 + 30 = 210

    Write the answer. = 150 or 210

    32

    -------

    13

    2-------

    2

    3

    4

    5

    2WORKEDExample

    yx--

    y

    x0

    0 or 360

    270

    180

    90

    1

    1

    1

    1

    MQ Maths B Yr 11 Ch 06 Page 240 Thursday, October 25, 2001 3:59 PM

  • C h a p t e r 6 Tr i g o n o m e t r i c e q u a t i o n s 241

    Simple trigonometric equations

    1 Solve each of the following trigonometric equations over the domain 0 360,correct to the nearest degree.

    2 Find exact solutions to each of the following trigonometric equations over the domain0 360.

    3

    If sin x = cos x = and 0 x 360, then x is:

    4 It is known that sin < 0 and that tan > 0. Which quadrant does the angle lie in?Explain your answer.

    a sin = 0.6 b cos = 0.25 c tan = 5.72 d sin = 0.85e cos = 0.195 f tan = 0.837 g sin = 0.333 h cos = 0.757

    a sin = b cos = c tan = d sin =

    e cos = f tan = g sin = h cos =

    A 150 or 210 B 135 or 225 C 225D 135 or 315 E 120

    Solve the equation sin = -1 in the domain 0 360.

    THINK WRITE

    Write the equation. sin = 1y = sin so find the angle with a y-value of 1.

    = 27012

    3WORKEDExample

    remember1. Trigonometric equations are equations that use the trigonometric ratios.2. The trigonometric functions are periodic and so they have an infinite number of

    solutions. The equation is usually written with a restricted domain to limit the number of answers.

    3. There are two solutions to most trigonometric equations with a domain0 360.

    4. Remember the special triangles as they are used in many solutions.5. Boundary angles may also provide the solution to an equation.

    remember

    6A

    SkillSH

    EET 6.1

    SkillSH

    EET 6.2

    WWORKEDORKEDEExample

    1

    WWORKEDORKEDEExample

    23

    2------- 2

    2------- 3

    12---

    12--- 3

    3------- 2

    2------- 3

    2-------

    mmultiple choiceultiple choice

    22

    -------

    MQ Maths B Yr 11 Ch 06 Page 241 Thursday, October 25, 2001 3:59 PM

  • 242 M a t h s Q u e s t M a t h s B Ye a r 1 1 f o r Q u e e n s l a n d5 Yvonne is doing a trigonometric problem that has reduced to the equation sin = 1.5.

    a When Yvonne tries to solve this equation her calculator returns an error message.Why?

    b When checking her working Yvonne realises that she should have used the tangentratio. Why is it now possible to achieve a solution to the equation tan = 1.5?

    6 Solve each of the following equations over the domain 0 360.

    7 Solve the following trigonometric equations over the domain 360 360.

    Equations using radiansWe have seen that a radian is an alternative method of measuring an angle. A trigon-ometric equation can be solved using radians as well as degrees. Usually the domaingiven will indicate whether it is expected that you will solve the equation in degrees orin radians.

    For example, if you are asked to solve an equation over the domain 0 360then degrees are expected for the answer. However, if the given domain is 0 2then it is expected that the answer will be given in radians.

    The method of solving the equations is the same, but be sure that your calculator isin radian mode before attempting to solve the problem to give an answer in radians.

    a sin = 1 b cos = 0 c tan = 0 d sin = 0e cos = 1 f sin = 1

    a sin = 0.5 b cos = 0.35 c tan = 1 d sin = 0.87e cos = 0.87 f tan = 1.4

    WWORKEDORKEDEExample

    3

    Solve the equation tan = 0.8 over the domain 0 2. Give the answer correct to 2 decimal places.

    THINK WRITE

    Write the equation. tan = 0.8Use your calculator to find the first quadrant angle.

    First quadrant angle = 0.67

    The tangent ratio is positive in the first and third quadrants.

    Find the third quadrant angle by adding 0.67 to .

    + 0.67 = 3.81

    Write the answer. = 0.67 or 3.81

    12

    y

    x0.67

    + 0.67

    3

    4

    5

    4WORKEDExample

    MQ Maths B Yr 11 Ch 06 Page 242 Thursday, October 25, 2001 3:59 PM

  • C h a p t e r 6 Tr i g o n o m e t r i c e q u a t i o n s 243When the special angles are used, it is still important to recognise them and recog-

    nise their radian equivalents in terms of .

    sin = cos = tan =

    sin = cos = tan = 1

    sin = cos = tan =

    All of the equations that we hav