Question

Find exact value of the expression?
tan a = 24/7, a lies in quadrant 3 and cos B =-15/17, B lies in quadrant 2. Find sin(a+B)

Answer 1

Answer on Question #81502 – Math – Trigonometry

Question

Find exact value of the expression?

tan a = 24/7, a lies in quadrant 3 and cos B = -15/17, B lies in quadrant 2. Find sin(a+B)

Solution

$$\sin(a+B) = \sin(a) \cdot \cos(B) + \cos(a) \cdot \sin(B),$$ $$\tan^2(a) = \sin^2(a) / \cos^2(a) = (1 - \cos^2(a) / \cos^2(a)) = 1 / \cos^2(a) - 1,$$ $$1 / \cos^2(a) = 1 + \tan^2(a),$$ $$\cos^2(a) = 1 / (1 + \tan^2(a)) = 1 / (1 + 24^2 / 7^2) = 1 / (625 / 49) = 49 / 625;$$ $$a \text{ lies in quadrant 3, hence } \cos(a) < 0, \sin(a) < 0,$$ $$\cos(a) = -\sqrt{\frac{49}{625}} = -7 / 25, \sin^2(a) = 1 - \cos^2(a) = 576 / 625, \quad \sin(a) = -\sqrt{\frac{576}{625}} = -24 / 25,$$ $$\sin^2(B) = 1 - \cos^2(B) = 1 - 225 / 289 = 64 / 289,$$ $$B \text{ lies in quadrant 2, hence } \sin(B) > 0,$$ $$\sin(B) = \sqrt{\frac{64}{289}} = 8 / 17,$$ $$\sin(a+B) = \sin(a) \cdot \cos(B) + \cos(a) \cdot \sin(B) = (-24 / 25) \cdot (-15 / 17) + (-7 / 25) \cdot (8 / 17) = (360 - 56) / 425 = 304 / 425$$

Answer: 304/425.

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