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Math Help - Compound Angle Formulas

  1. #1
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    Compound Angle Formulas

    Can someone describe to me clearly, how you would solve these questions:

    1. Angles x and y are located in the first quadrant such that sinx=4/5 and cosy=7/25.
    a) determine an exact value for cos x
    b) determine an exact value for sin y
    c) determine an exact value for sin(x+y)

    2. angle x lies in the third quadrant, and tanx=7/24
    a) determine an exact value for cos2x
    b) determine an exact value for sin2x

    for 1 a and b I got the answers, but for 1 c, I am not sure how to approach it
    Last edited by skeske1234; March 28th 2009 at 04:27 PM.
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  2. #2
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    1) for sin(x) = 4/5
    you can construct the triangle with opposite length 4 and hypotenuse 5. you can find angle x using sin^-1. then find the adjacent side and find cos(x).
    use similar methods for part b. use double angle forumla for part c.

    2) construct the triangle like you did for part 1, then solve
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  3. #3
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    for part c of question 1, if I get to
    sin(4/5)cos(7/25)+cos(3/5)sin(24/25)
    what do I do next?
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  4. #4
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    Trig Ratios

    Hello skeske
    Quote Originally Posted by skeske1234 View Post
    Can someone describe to me clearly, how you would solve these questions:

    1. Angles x and y are located in the first quadrant such that sinx=4/5 and cosy=7/25.
    a) determine an exact value for cos x
    b) determine an exact value for sin y
    c) determine an exact value for sin(x+y)
    This is what you need to know to solve a question like this:

    • If an angle is in the first quadrant, then its sine and cosine are both positive.
    • The relation between sine and cosine is \sin^2x+\cos^2x=1

    So: \sin x =\frac45 \Rightarrow \left(\frac45\right)^2 +\cos^2x = 1

     \Rightarrow \cos^2x = 1 - \frac{16}{25}=\frac{9}{25}

    \Rightarrow \cos x = \frac{3}{5}, since \cos x > 0

    In the same way, if \cos y = \frac{7}{25}, \sin y = \frac{24}{25}
    Quote Originally Posted by skeske1234 View Post
    for part c of question 1, if I get to
    sin(4/5)cos(7/25)+cos(3/5)sin(24/25)
    what do I do next?
    You're getting things a bit confused here.

    \sin(x+y) =\sin x\cos y + \cos x \sin y = \frac{4}{5}\cdot \frac{7}{25} + \frac{3}{5}\cdot \frac{24}{25} = \frac{28}{125}+ \frac{72}{125}= \frac{100}{125} = \frac{4}{5}

    2. angle x lies in the third quadrant, and tanx=7/24
    a) determine an exact value for cos2x
    b) determine an exact value for sin2x
    If angle x lies in the third quadrant, then \sin x and \cos x are both negative.

    If you know the value of \tan x, then use:

    \sec^2 x = 1 + \tan^2x

    and then use \cos^2x = \frac{1}{\sec^2 x}

    Then, \cos 2x = 2\cos^2x - 1

    Finally, for \sin 2x, the quickest way is to use \sin^22x = 1 - \cos^22x

    Note that \sin 2x will be positive, since \sin x and \cos x are both negative and \sin 2x = 2 \sin x \cos x

    Can you complete this? If not, let's see your working, and we'll finish it for you.

    Grandad
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