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Thread: trigonometry equatioonss partt 2 =(

  1. #1
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    trigonometry equatioonss partt 2 =(

    hii guyzz, having trouble with MORE of these trig equations ><""

    use the factors of xcubed - ycubed to show that
    cos^6x - sin^6x = (1- 1/4sin^2x)cos2x

    by the way, this ^means to the power to, and its just to the power of 6, and NOT to the power of 6x if anyone gets confued, the x is like theta

    AND

    if cosx = (a^2 - m^2)/(a^2 + m^2) and 0<x<pi/2 express tanx and sin2x in terms of a and m

    thankyou sooooo much to anyone who can give me some help in these equations, i'm so grateful =D
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  2. #2
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    Quote Originally Posted by iiharthero View Post
    hii guyzz, having trouble with MORE of these trig equations ><""

    use the factors of xcubed - ycubed to show that
    cos^6x - sin^6x = (1- 1/4sin^2x)cos2x

    by the way, this ^means to the power to, and its just to the power of 6, and NOT to the power of 6x if anyone gets confued, the x is like theta
    Hi

    You are told to use $\displaystyle a^3 - b^3 = (a-b)(a^2+ab+b^2)$ with $\displaystyle a = \cos^2 x$ and $\displaystyle b = \sin^2 x$
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  3. #3
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    Quote Originally Posted by iiharthero View Post
    if cosx = (a^2 - m^2)/(a^2 + m^2) and 0<x<pi/2 express tanx and sin2x in terms of a and m
    $\displaystyle \cos x = \frac{a^2 - m^2}{a^2 + m^2}$
    cos x is also adjacent over hypotenuse in a right triangle, so let's say that the adjacent side is $\displaystyle a^2 - m^2$ and that the hypotenuse is $\displaystyle a^2 + m^2$.

    Use the Pythagorean theorem to find the remaining (opposite) side:
    $\displaystyle \begin{aligned}
    (a^2 - m^2)^2 + y^2 &= (a^2 + m^2)^2 \\
    a^4 - 2a^2m^2 + m^4 + y^2 &= a^4 + 2a^2m^2 + m^4 \\
    -2a^2m^2 + y^2 &= 2a^2m^2 \\
    y^2 &= 4a^2m^2 \\
    y &= \pm 2am
    \end{aligned}$

    Since $\displaystyle 0 < x < \frac{\pi}{2}$, we can drop the $\displaystyle \pm$ sign. y = 2am

    So,
    adjacent side is $\displaystyle a^2 - m^2$,
    opposite side is $\displaystyle 2am$, and
    hypotenuse is $\displaystyle a^2 + m^2$.

    Use $\displaystyle \sin x = \frac{\text{opp}}{\text{hyp}}$ to find sin x.

    Use $\displaystyle \tan x = \frac{\text{opp}}{\text{adj}}$ or $\displaystyle \frac{\sin x}{\cos x}$ to find tan x.

    Use $\displaystyle \sin 2x = 2\sin x \cos x$ to find sin 2x.


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