Answer to Question #167790 in Civil and Environmental Engineering for eyel

Question

1.      Given that sin α =  and sin β = , find 𝑡𝑎𝑛 (α + β) if both α and β are in QIV.

Answer 1

$$\sin \alpha=a, -1\leq a\leq0$$

$$\sin\beta=b, -1\leq a\leq0$$

$$\cos \alpha=\sqrt{1-\sin^2 \alpha}=\sqrt{1-a^2}$$

$$\cos\beta =\sqrt{1-\sin^2 \beta}=\sqrt{1-b^2}$$

$$\tan(\alpha+\beta)=\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$$

$$=\dfrac{\dfrac{a}{\sqrt{1-a^2}}+\dfrac{b}{\sqrt{1-b^2}}}{1-\dfrac{a}{\sqrt{1-a^2}}\dfrac{b}{\sqrt{1-b^2}}}$$

$$=\dfrac{a\sqrt{1-b^2}+b\sqrt{1-a^2}}{\sqrt{1-a^2}\sqrt{1-b^2}-ab}$$

$$\tan(\alpha+\beta)=\dfrac{a\sqrt{1-b^2}+b\sqrt{1-a^2}}{\sqrt{1-a^2}\sqrt{1-b^2}-ab}$$