Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

Trigonometry Help ASAP ?

Given sinx = 2/3 and cos < 0

a) Use double identities to find the exact values of sin2x and cos2x

    I got cos2x =1/9 and sin2x =4\sqrt(5)/9  

    I'm just confused on how to do b & c.

b) Use a Pythagorean Identity to verify the answers you found for sin2x and cos2x.

c) Use a Ratio Identity to find tan2x. Do not use the double angel identity for tangent. 

2 Answers

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  • 2 months ago

    sinx = 2/3 and cosx < 0...hence, x is in Qii

    a) sin2x = 2sinxcosx

    Now, sin²x + cos²x = 1

    so, cos²x = 1 - sin²x

    i.e. cos²x = 1 - (2/3)² => 5/9

    Hence, cosx = ±√5/3

    Then, cosx = -√5/3...as x is in Qii

    so, sin2x = 2(2/3)(-√5/3) => -4√5/9 

    Note: your result of 4\√5/9...is not correct

    Now, cos2x = 1 - 2sin²x

    so, cos2x = 1 - 2(2/3)² => 1/9

    Note: cos2x > 0 becuase if x is in Qii, then 2x is in Qiv where cos2x > 0

    b) We already know that sin²x + cos²x = 1

    so, sin²2x + cos²2x = 1

    i.e. (-4√5/9)² + (1/9)² => 80/81 + 1/81 = 1

    c) tan2x = sin2x/cos2x => (-4√5/9)/(1/9)

    i.e. tan2x = -4√5

    Note: if x is in Qii, then 2x is in Qiv where tan2x < 0

    :)>

  • ted s
    Lv 7
    2 months ago

    so the reference triangle  for x is { √5 , 2 , 3 } ===> cos 2x = 5/9 - 4 / 9 = 1 / 9 ====> reference triangle for 2x is { 1 , √80 , 9 } ===> sin 2x = √80 / 9 and tan 2x =  √80....b) & c) contained in this work

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