Trigonometry

4. Find the exact value for sin(x+y) if sinx=-4/5 and cos y = 15/17. Angles x and y are in the fourth quadrant.

5. Find the exact value for cos 165degrees using the half-angle identity.

1. Solve: 2 cos^2x - 3 cosx + 1 = 0 for 0 less than or equal to x <2pi.

2. Solve: 2 sinx - 1 = 0 for 0degrees less than or equal to x <360degrees.

3. Solve: sin^2x = cos^2x for 0degrees is less than or equal to x < 360degrees.

4. Solve sinx - 2sinx cosx = 0 for 0 is less than or equal to x < 2pi.

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  1. #4
    sin(x+y) = sinx cosy + cosx siny

    so we need the cosx and the siny
    from sinx = -4/5 we know we are dealing with the 3,4,5 right-angled triangle, so in the fourth quadrant cosx = 3/5
    from cosy = 15/17 we know we are dealing with the 8,15,17 right-angled triangle
    so in the fourth quadrant, siny = -8/17
    then
    sin(x+y) = sinx cosy + cosx siny
    = (-4/5)(15/17) + (3/5)(-8/17)
    = -84/85

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  2. cos 2A = 2cos^2 A - 1
    let A = 165, then 2A = 330
    so let's find cos 330
    cos(330)
    = cos(360-30)
    = cos360 cos30 + sin360 sin30
    = (1)(√3/2) + (0)(1/2) = √3/2

    then √3/2 = 2cos^2 165 - 1
    (√3 + 2)/4 = cos^2 165
    cos 165 = -√(√3 + 2))/2

    (algebraically, our answer would have been ± , but I picked the negative answer since 165 is in the second quadrant, and the cosine is negative in the second quadrant)

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  3. I will give you hints for the rest,
    you do them, and let me know what you get.

    #1 factor it as
    (2cosx - 1)(cosx -1) = 0
    so cosx - 1/2 or cosx = 1
    take it from there, you should get 3 answers.

    #2, the easiest one
    take the 1 to the other side, then divide by 2,

    #3, take √ of both sides to get
    sin 2x = ± cos 2x
    sin2x/cos2x = ± 1
    tan 2x = ± 1 etc

    #4 factor out a sinx etc

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