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Math Help - Stumped... cosθ=sin(90-θ)

  1. #1
    Member wiseguy's Avatar
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    Stumped... cosθ=sin(90-θ)

    "Why is cosθ=sin(90-θ)? Use the figure to help explain."


    I attached the image.


    In this instance, I am at a complete loss...

    ??
    Attached Thumbnails Attached Thumbnails Stumped... cosθ=sin(90-θ)-t15_6.jpg  
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  2. #2
    A Plied Mathematician
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    Drop a perpendicular from the point (u,v) down to the u axis, and another perpendicular from (u,v) horizontally over to the v axis. Label the line from the origin to (u,v) - call it h. Now, you tell me: from the definitions, what is the cosine of theta? What is the sin of (90 - theta)?
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  3. #3
    Member Mathelogician's Avatar
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    Make right triangles from the point (u,v) and use the definitions of sine and cosine. Or use a unit circle for another geometric proof.
    But it's simpler to use the sine Difference Formula!
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  4. #4
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    Quote Originally Posted by wiseguy View Post
    "Why is cosθ=sin(90-θ)? Use the figure to help explain."


    I attached the image.


    In this instance, I am at a complete loss...

    ??
    For simplicity, if you draw a single right-angled triangle.

    One acute angle is \theta and the other is 90^o-\theta since the acute angles sum to 90 degrees in a right-angled triangle.

    Place \theta at the acute angle coming away from the base.

    Label that side at \theta "base"

    Then cos\theta=\frac{base}{hypotenuse} as the base is adjacent \theta

    sin\left(90^o-\theta\right)=\frac{base}{hypotenuse} as the base is opposite the angle 90^o-\theta
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