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Math Help - Finding the exact value of Inverse Trig Functions without a calculator

  1. #1
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    Finding the exact value of Inverse Trig Functions without a calculator

    Hey I'm in 11th grade in Advanced Trigonometry and need help with Inverse Trig Functions,

    Find the exact value of each expression without calculator.

    sin [arctan (-5/12)]

    cot [arccsc(-9/4)]

    -Those are examples of the problems I need to solve any help would be greatly appreciated

    Thanks,
    Sean
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  2. #2
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    Re: Finding the exact value of Inverse Trig Functions without a calculator

    If \theta= arctan(-5/12) then tan(\theta)= -5/12= "opposite side"/"near side'. Draw a right triangle with one vertex at (0, 0), another at (-12, 0), and the third at (-12, 5). Label the angle at the origin " \theta". On that triangle tan(\theta)= -5/12. What is sin(\theta)?

    If \theta =arccsc(-9/4), then [tex]csc(\theta)= -9/4[//tex]= "hypotenuse"/"opposite side". In order to draw such a triangle, we need to know the length of the "near side". Use the Pythagorean theorem.
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    Re: Finding the exact value of Inverse Trig Functions without a calculator

    Quote Originally Posted by srob22 View Post
    Hey I'm in 11th grade in Advanced Trigonometry and need help with Inverse Trig Functions,

    Find the exact value of each expression without calculator.

    sin [arctan (-5/12)]

    cot [arccsc(-9/4)]

    -Those are examples of the problems I need to solve any help would be greatly appreciated

    Thanks,
    Sean
    What is the 3rd side in the right triangle with sides -5 and 12? -9 and 4? Identify the angle arctan and arccsc refer to and calculate the ratios for sin and cot in each respective triangle.
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    Re: Finding the exact value of Inverse Trig Functions without a calculator

    I missed the previous replies, but here goes.

    Quote Originally Posted by srob22 View Post
    sin [arctan (-5/12)]
    Draw the unit circle and an angle whose tangent is -5/12. Find the right triangle with hypotenuse 1 that is similar to the right triangle with legs 5 and 12. The smaller of those legs is the absolute value of sin [arctan (-5/12)]. For the sign, consider which of the two angles with tangent -5/12 is in the domain of arctangent.

    Alternatively, express sine through tangent using these formulas.

    Quote Originally Posted by srob22 View Post
    cot [arccsc(-9/4)]
    First, arccsc(x) = arcsin(1/x) and arcsin(-x) = -arcsin(x). Then again, either find the right triangle with hypotenuse 1 and one leg 4/9 and find the ratio of the legs, or express cotangent through sine.
    Last edited by emakarov; March 17th 2013 at 03:19 PM.
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    Re: Finding the exact value of Inverse Trig Functions without a calculator

    Quote Originally Posted by HallsofIvy View Post
    If \theta= arctan(-5/12) then tan(\theta)= -5/12= "opposite side"/"near side'. Draw a right triangle with one vertex at (0, 0), another at (-12, 0), and the third at (-12, 5). Label the angle at the origin " \theta". On that triangle tan(\theta)= -5/12. What is sin(\theta)?
    Yes, though the final result should be negative.
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