cos alpha equals 7 over 25 and sin beta equals 4 over 5. Both angles are in Quadrant I. Find cos left parenthesis alpha minus beta right par

Question

cos alpha equals 7 over 25 and sin beta equals 4 over 5. Both angles are in Quadrant I. Find cos left parenthesis alpha minus beta right parenthesis

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Hadley 2 weeks 2021-09-10T18:58:19+00:00 1 Answer 0

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    2021-09-10T19:00:18+00:00

    Answer:

    \dfrac{117}{125}

    Step-by-step explanation:

    We are given that:

    cos\alpha  = \dfrac{7}{25}

    sin \beta = \dfrac{4}{5}

    To find cos(\alpha - \beta).

    As per Formula:

    cos(A-B) =cos A cosB+ sinA sinB

    Here, A is \alpha and B is \beta.

    So, formula becomes

    cos(\alpha -\beta)=cos\apha cos\beta+sin\alpha sin \beta   ..... (1)

    Using the following identity to calculate sin\alpha \text{ and } cos \beta:

    sin^2 \theta + cos^2 \theta = 1

    sin^{2}\alpha+\dfrac {7^{2} }{25^{2} } = 1\\ \Rightarrow sin\alpha = \sqrt{1-\dfrac{49}{625}}\\\Rightarrow sin\alpha = \dfrac{24}{25}

    Similarly,

    \dfrac {4^{2} }{5^{2} } + cos^{2}\beta = 1\\ \Rightarrow cos\beta = \sqrt{1-\dfrac{16}{25}}\\\Rightarrow cos\beta = \dfrac{3}{5}

    Putting values in equation (1):

    cos(\alpha -\beta ) = \dfrac{7}{25} \times \dfrac{3}{5} + \dfrac{24}{25} \times \dfrac{4}{5}\\\Rightarrow \dfrac{21+96}{125}\\\Rightarrow \dfrac{117}{125}

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45:7+7-4:2-5:5*4+35:2 =? ( )