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# trig problem

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A and B are angles in the interval 0 < A, B < 45 degrees.  If cos(A + B) = 4/5 and sin(A - B) = 5/13, find tan 2A.

Jul 30, 2020

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A and B are angles in the interval $$0 < A, B < 45^\circ$$.
If $$\cos(A + B) = \dfrac{4}{5}$$ and $$\sin(A - B) = \dfrac{5}{13}$$, find $$\tan(2A)$$.

$$\begin{array}{|rcll|} \hline \cos(A + B) &=& \dfrac{4}{5} \\ A + B &=& \arccos\left( \dfrac{4}{5} \right) \qquad (1) \\ \hline \end{array} \begin{array}{|rcll|} \hline \sin(A - B) &=& \dfrac{5}{13} \\ A - B &=& \arcsin\left( \dfrac{5}{13} \right) \qquad (2) \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline (A + B) +(A - B) &=& \arccos\left( \dfrac{4}{5} \right)+\arcsin\left( \dfrac{5}{13} \right) \\ 2A &=& \arccos\left( \dfrac{4}{5} \right)+\arcsin\left( \dfrac{5}{13} \right) \\ \tan(2A) &=& \tan\Bigg( \arccos\left( \dfrac{4}{5} \right)+\arcsin\left( \dfrac{5}{13} \right) \Bigg) \\ \tan(2A) &=& \tan\Bigg( 36.8698976458^\circ+22.6198649480^\circ \Bigg) \\ \tan(2A) &=& \tan\Bigg( 59.4897625939^\circ \Bigg) \\ \tan(2A) &=& 1.69696969697 \\\\ \mathbf{\tan(2A)} &=& \mathbf{\dfrac{56}{33}} \\ \hline \tan(2A) &=& \tan\Bigg( 59.4897625939^\circ \Bigg) \\ 2A &=& 59.4897625939^\circ \\ \mathbf{A} &=& \mathbf{29.7448812969^\circ} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline (A + B) -(A - B) &=& \arccos\left( \dfrac{4}{5} \right)-\arcsin\left( \dfrac{5}{13} \right) \\ 2B &=& \arccos\left( \dfrac{4}{5} \right)-\arcsin\left( \dfrac{5}{13} \right) \\ \tan(2B) &=& \tan\Bigg( \arccos\left( \dfrac{4}{5} \right)-\arcsin\left( \dfrac{5}{13} \right) \Bigg) \\ \tan(2B) &=& \tan\Bigg( 36.8698976458^\circ-22.6198649480^\circ \Bigg) \\ \tan(2B) &=& \tan\Bigg( 14.2500326978^\circ \Bigg) \\ \tan(2B) &=& 0.25396825397 \\\\ \mathbf{\tan(2B)} &=& \mathbf{\dfrac{16}{63}} \\ \hline \tan(2B) &=& \tan\Bigg( 14.2500326978^\circ \Bigg) \\ 2B &=& 14.2500326978^\circ \\ \mathbf{B} &=& \mathbf{7.12501634890^\circ} \\ \hline \end{array}$$

Jul 30, 2020