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Math Help - Due Tomorrow! Solving Trig equations

  1. #1
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    Exclamation Due Tomorrow! Solving Trig equations

    Hey, I'm stuck on these 5 problems and can't seem to figure out what's what. I'm supposed to express in terms of *blah* + 2k(pi) i believe. Any help appreciated. I'll continue working on them and check back later. I really need the points for these, so I kinda need to get em all right ^_^

    1. Find all solutions of the following. Express in radians.

    a. 2cos^2(x-3)cos(x+1) = 0

    b. 4tan(3x - 4) = 0

    c. sin(2x - 1) = -1/2

    2. Find the exact solutions of sin(x+2)csc(x) = 3 on the interval [0, 2(pi)]. Express in radians.

    3. Find all solutions of csc(x) + cot(x) = sqrt(3). express answer in radians. Check answers. Hint: rewrite in terms of sine and cosine, clear fractions by multiplying through by the denominator, and square both sides.
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  2. #2
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    Hello, leviathanwave!

    Here's some help . . .


    1. Find all solutions of the following. .Express in radians.

    a. .2·cos²(x - 3)·cos(x + 1) .= .0

    Set each factor equal to zero.

    2·cos²(x - 3) = 0 . . cos(x - 3) = 0 . . x - 3 .= .½π + 2kπ . . x .= .½π + 3 + 2kπ

    cos(x + 1) = 0 . . x + 1 .= .±½π + 2kπ . . x .= . ±½π - 1 + 2kπ



    b. .4·tan(3x - 4) .= .0

    We have: .tan(3x - 4) .= .0 . . 3x - 4 .= .0 + kπ . . 3x .= .4 + kπ

    Therefore: .x .= .(4/3) + (kπ/3)



    c. .sin(2x - 1) .= .

    We have: .sin(2x - 1) .= . . . 2x - 1 .= .arcsin(-½)

    . . which has "two" values: .(7π/6) + 2kπ .and .(11π/6) + 2kπ


    2x - 1 .= .(7π/6) + 2kπ . . 2x .= .(7π/6) + 1 + 2kπ . . x .= .(7π/12) + ½ + kπ

    2x - 1 .= .(11π/6) + 2kπ . . 2x .= .(11π/6) + 1 + 2kπ . . x .= .(11π/12) + ½ + kπ

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  3. #3
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    Exclamation thanks!

    Anyone have any ideas on the last two? specifically the last one.
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  4. #4
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    Quote Originally Posted by leviathanwave View Post
    Hey, I'm stuck on these 5 problems and can't seem to figure out what's what. I'm supposed to express in terms of *blah* + 2k(pi) i believe. Any help appreciated. I'll continue working on them and check back later. I really need the points for these, so I kinda need to get em all right ^_^

    1. Find all solutions of the following. Express in radians.

    a. 2cos^2(x-3)cos(x+1) = 0

    b. 4tan(3x - 4) = 0

    c. sin(2x - 1) = -1/2

    2. Find the exact solutions of sin(x+2)csc(x) = 3 on the interval [0, 2(pi)]. Express in radians.

    3. Find all solutions of csc(x) + cot(x) = sqrt(3). express answer in radians. Check answers. Hint: rewrite in terms of sine and cosine, clear fractions by multiplying through by the denominator, and square both sides.

    Okay, this is for #3:

    Things to keep aware of:
    csc(x) = 1/sin(x)
    cot(x) = 1/tan(x) = cos(x)/sin(x)

    csc(x) + cot(x) = sqrt(3)
    1/sin(x) + cos(x)/sin(x) = sqrt(3)

    Multiply by sin(x)

    1 + cos(x) = sqrt(3)sin(x)

    Square both sides

    1 + 2cos(x) + cos^2(x) = 3sin^2(x)

    Now using sin^2(x) = (1 - cos^2(x)):

    1 + 2cos(x) + cos^2(x) = 3(1 - cos^2(x))
    1 + 2cos(x) + cos^2(x) = 3 - 3cos^2(x)
    4cos^2(x) + 2cos(x) - 2 = 0
    (4cos(x) - 2)(cos(x) + 1) = 0

    Now solve for each:
    4cos(x) - 2 = 0
    4cos(x) = 2
    cos(x) = 1/2
    x = pi/3, 5pi/3

    cos(x) + 1 = 0
    cos(x) = -1
    x = pi

    Solutions in ascending order:
    pi/3
    pi
    5pi/3

    Hope this helps! If I can do #2, I'll post in a few minutes.
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  5. #5
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    Exclamation thanks!

    Yeah number 2 is a tough one, i'm still lookin for help if anyone can
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  6. #6
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    Quote Originally Posted by leviathanwave View Post
    2. Find the exact solutions of sin(x+2)csc(x) = 3 on the interval [0, 2(pi)]. Express in radians.
    sin(x + 2) = sin(x)cos(2) + sin(2)cos(x)
    csc(x) = 1/sin(x)

    So
    sin(x + 2)csc(x) = cos(2) + sin(2)cot(x) = 3

    Thus
    cot(x) = (3 - cos(2))/sin(2)

    tan(x) = sin(2)/(3 - cos(2))

    I can't think of a way to get an exact solution set for this so I will merely state:
    x = 1.25935 rad and x = 4.40094 rad

    -Dan
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