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Math Help - Trigonometry surds

  1. #1
    RAz
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    Trigonometry surds

    If sinA=\frac{2}{\sqrt{5}} and tanB=\frac{1}{2}, how can I prove that A-B=45?

    Using a calculator to find inverse sine and tangent then subtracting the angles gets 45 degrees, but I do not think that is the correct way to approach the question.

    Similar questions are sin(A+B+C)=\frac{2\sqrt{2}}{3} find sinC etc.

    I think when I get the method I should be fine in doing these questions. Thanks!
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  2. #2
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    Quote Originally Posted by RAz View Post
    If sinA=\frac{2}{\sqrt{5}} and tanB=\frac{1}{2}, how can I prove that A-B=45?

    Using a calculator to find inverse sine and tangent then subtracting the angles gets 45 degrees, but I do not think that is the correct way to approach the question.

    Similar questions are sin(A+B+C)=\frac{2\sqrt{2}}{3} find sinC etc.

    I think when I get the method I should be fine in doing these questions. Thanks!
    sinA = \frac{2}{\sqrt{5}}

    CosA = \sqrt{1-sin^2(A)}

    Then find tanA.

    Now tan(A-B) = \frac{tanA - tanB}{1+tanAtanB}

    Substitute the values of tanA and tanB and simplify.
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  3. #3
    MHF Contributor Unknown008's Avatar
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    You might try using simple sketches.

    sin A = \frac{2}{\sqrt{5}}

    Then, draw a simple right angled triangle, with hypotenuse [/tex]\sqrt5[/tex], opposite side 2 and adjacent side 1.

    Ah, coincidence, the angle opposite to the angle A is the angle B!

    Now that you know all the trigonometric ratios of A and B, you can use any identity you know;

    sin(A-B) = sinAcosA - sinBcosB

    cos(A-B) = cosAcosB + sinAsinB

    or the tangent ratio given by sa-ri-ga-ma.

    For cos and sin, you should get cos(A-B) = sin(A-B) = \frac{1}{\sqrt2}

    For tan, you should get tan(A-B) = 1.

    ~~~~~~~~~

    For the second question, provided you know A, B, or one trigonometric identity of A and B, you use the identity:

    sin(A+B) = sinAcosA + sinBcosB

    Where, you replace A by (A+B) and B by C.

    Then, you can use the identity : sin(2A) = 2 sinA cos A to solve for C.
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