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Question: If sin α + sin β = 1/3 and cos α + cos β = 1/4. Then the value of sin(α + β) is...

If sin α + sin β = 1/3 and cos α + cos β = 1/4. Then the value of sin(α + β) is

Answer

24/25

Explanation

Solution

Solution:

Given

sin⁡α+sin⁡β=13andcos⁡α+cos⁡β=14.\sin \alpha+\sin \beta=\frac{1}{3} \quad \text{and} \quad \cos \alpha+\cos \beta=\frac{1}{4}.

  1. Express the sums using sum-to-product formulas:

    sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}

  2. Divide the two equations:

    sin⁡α+sin⁡βcos⁡α+cos⁡β=2sin⁡α+β2cos⁡α−β22cos⁡α+β2cos⁡α−β2=tan⁡α+β2\frac{\sin \alpha+\sin \beta}{\cos \alpha+\cos \beta}=\frac{2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}}{2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}}=\tan\frac{\alpha+\beta}{2}

    So,

    tan⁡α+β2=1314=43\tan\frac{\alpha+\beta}{2}=\frac{\frac{1}{3}}{\frac{1}{4}}=\frac{4}{3}

  3. For tan⁡θ=43\tan\theta=\frac{4}{3}, consider a right triangle with opposite side 4, adjacent side 3. Then,

    sin⁡α+β2=45,cos⁡α+β2=35.\sin\frac{\alpha+\beta}{2}=\frac{4}{5},\quad \cos\frac{\alpha+\beta}{2}=\frac{3}{5}.

  4. Use the double angle formula:

    sin⁡(α+β)=2sin⁡α+β2cos⁡α+β2=2(45)(35)=2425.\sin(\alpha+\beta)=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha+\beta}{2}=2\left(\frac{4}{5}\right)\left(\frac{3}{5}\right)=\frac{24}{25}.