Determine A and A from A = (4)/(3).

Mathematics

ScanSolve AI Answer

Answer

B

Step-by-step solution

3 steps

Question 1: If tan⁡A=43\tan A = \frac{4}{3} and cos⁡B=513\cos B = \frac{5}{13}, where both AA and BB are acute angles, what is the exact value of sin⁡(A+B)\sin(A + B)?

  1. Determine sin⁡A\sin A and cos⁡A\cos A from tan⁡A=43\tan A = \frac{4}{3}.

    Since AA is an acute angle, we can use a right-angled triangle.
    Opposite side = 4, Adjacent side = 3.
    Hypotenuse = 42+32=16+9=25=5\sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5.
    Therefore, sin⁡A=OppositeHypotenuse=45\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{5} and cos⁡A=AdjacentHypotenuse=35\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{3}{5}.

  2. Determine sin⁡B\sin B from cos⁡B=513\cos B = \frac{5}{13}.

    Since BB is an acute angle, we can use the identity sin⁡2B+cos⁡2B=1\sin^2 B + \cos^2 B = 1.

    sin⁡2B+(513)2=1\sin^2 B + \left(\frac{5}{13}\right)^2 = 1 sin⁡2B+25169=1\sin^2 B + \frac{25}{169} = 1 sin⁡2B=1−25169=169−25169=144169\sin^2 B = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169}

    Since BB is acute, sin⁡B\sin B is positive.

    sin⁡B=144169=1213\sin B = \sqrt{\frac{144}{169}} = \frac{12}{13}
  3. Use the sum formula for sine, sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A \cos B + \cos A \sin B.

    Substitute the values found:

    sin⁡(A+B)=(45)(513)+(35)(1213)\sin(A+B) = \left(\frac{4}{5}\right)\left(\frac{5}{13}\right) + \left(\frac{3}{5}\right)\left(\frac{12}{13}\right) sin⁡(A+B)=2065+3665\sin(A+B) = \frac{20}{65} + \frac{36}{65} sin⁡(A+B)=20+3665\sin(A+B) = \frac{20 + 36}{65} sin⁡(A+B)=5665\sin(A+B) = \frac{56}{65}

    Comparing this with the given options, the correct option is (b).

    The exact value of sin⁡(A+B)\sin(A + B) is (b) 5665\boxed{\text{(b) } \frac{56}{65}}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question