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

Question #167790

1.      Given that sin Ξ± =  and sin Ξ² = , find π‘‘π‘Žπ‘› (Ξ± + Ξ²) if both Ξ± and Ξ² are in QIV.


1
Expert's answer
2021-03-04T06:02:51-0500


sin⁑α=a,βˆ’1≀a≀0\sin \alpha=a, -1\leq a\leq0

sin⁑β=b,βˆ’1≀a≀0\sin\beta=b, -1\leq a\leq0

cos⁑α=1βˆ’sin⁑2Ξ±=1βˆ’a2\cos \alpha=\sqrt{1-\sin^2 \alpha}=\sqrt{1-a^2}

cos⁑β=1βˆ’sin⁑2Ξ²=1βˆ’b2\cos\beta =\sqrt{1-\sin^2 \beta}=\sqrt{1-b^2}


tan⁑(Ξ±+Ξ²)=tan⁑α+tan⁑β1βˆ’tan⁑αtan⁑β\tan(\alpha+\beta)=\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}

=a1βˆ’a2+b1βˆ’b21βˆ’a1βˆ’a2b1βˆ’b2=\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}}}

=a1βˆ’b2+b1βˆ’a21βˆ’a21βˆ’b2βˆ’ab=\dfrac{a\sqrt{1-b^2}+b\sqrt{1-a^2}}{\sqrt{1-a^2}\sqrt{1-b^2}-ab}

tan⁑(Ξ±+Ξ²)=a1βˆ’b2+b1βˆ’a21βˆ’a21βˆ’b2βˆ’ab\tan(\alpha+\beta)=\dfrac{a\sqrt{1-b^2}+b\sqrt{1-a^2}}{\sqrt{1-a^2}\sqrt{1-b^2}-ab}


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