Question #71326

Find the exact values of the trigonometric functions of a/2 and 2a by using half-angle and double angle formulas.



1) sin(a) = 4/5, and a in quadrant 1


1
Expert's answer
2021-11-01T18:44:53-0400

1) Use half-angle identity for sine:


sina2=±1cosa2sin \frac{a}{2}=±\sqrt \frac{1-cos a}{2}

Find cosa:cos a:


cosa=±1sin2acos a=±\sqrt{1-sin^2 a}cosa=±1(45)2=±11625=±925=±35cos a=±\sqrt{1-(\frac{4}{5})^2}=±\sqrt{1-\frac{16}{25}}=±\sqrt{\frac{9}{25}}=±\frac{3}{5}

Since aa is in the quadrant I, cosa>0,cosa=35cos a>0, cos a=\frac{3}{5}

Hence,


sina2=±1352=±252=±15sin \frac{a}{2}=±\sqrt{\frac{1-\frac{3}{5}}{2}}=±\sqrt{\frac{\frac{2}{5}}{2}}=±\sqrt{\frac{1}{5}}

Since aa is in the quadrant I, sina2>0,sina2=15sin \frac{a}{2}>0, sin \frac{a}{2}=\sqrt{\frac{1}{5}}

2) Use half-angle identity for cosine:


cosa2=±1+cosa2cos \frac{a}{2}=±\sqrt \frac{1+cos a}{2}cosa2=±1+352=±852=±45cos \frac{a}{2}=±\sqrt{\frac{1+\frac{3}{5}}{2}}=±\sqrt{\frac{\frac{8}{5}}{2}}=±\sqrt{\frac{4}{5}}


Since aa is in the quadrant I, cosa2>0,cosa2=45cos \frac{a}{2}>0, cos \frac{a}{2}=\sqrt{\frac{4}{5}}

3) Half-angle for tangent and cotangent:


tana2=sina2:cosa2=15:45=14=12tan \frac{a}{2}=sin \frac{a}{2}:cos \frac{a}{2}=\sqrt{\frac{1}{5}}:\sqrt{\frac{4}{5}}=\sqrt{\frac{1}{4}}=\frac{1}{2}cota2=cosa2:sina2=45:15=41=2cot \frac{a}{2}=cos \frac{a}{2}:sin \frac{a}{2}=\sqrt{\frac{4}{5}}:\sqrt{\frac{1}{5}}=\sqrt{\frac{4}{1}}=2

4) Use double-angle identity for sine:


sin2a=2sinacosasin 2a=2sinacos asin2a=24535=2425sin 2a=2\cdot \frac{4}{5} \cdot \frac{3}{5}=\frac{24}{25}

5) Use double-angle identity for cosine:


cos2a=12sin2a=12(45)2=13225=725cos 2a=1-2sin^2a=1-2\cdot({\frac{4}{5}})^2=1-\frac{32}{25}=-\frac{7}{25}

6) Double-angle for tangent and cotangent:


tan2a=sin2a:cos2a=2425:(725)=247tan2a=sin2a:cos 2a=\frac{24}{25}:(-\frac{7}{25})=-\frac{24}{7}

cot2a=cos2a:sin2a=725:2425=724cot2a=cos2a:sin 2a=-\frac{7}{25}:\frac{24}{25}=-\frac{7}{24}

Answer: sina2=15,cosa2=45,tana2=12,cota2=2,sin \frac{a}{2}=\sqrt{\frac{1}{5}}, cos \frac{a}{2}=\sqrt{\frac{4}{5}}, tan \frac{a}{2}=\frac{1}{2}, cot\frac{a}{2}=2,

sin2a=2425,cos2a=725,tan2a=247,cot2a=724sin 2a=\frac{24}{25}, cos 2a=-\frac{7}{25}, tan 2a=-\frac{24}{7}, cot 2a=-\frac{7}{24}


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