Answer:
Alpha is 54 degrees
Step-by-step explanation:
We can use the addition identities for cosine and for sine to express the given equation in terms of sine and cosine of alpha:
[tex]cos(3\alpha)= - sin(2\alpha)\\cos(2\alpha)*cos(\alpha)-sin(2\alpha)*sin(\alpha) = - 2*sin(\alpha)*cos(\alpha)\\(cos^2(\alpha)-sin^2(\alpha))*cos(\alpha)-2*sin^2(\alpha)*cos(\alpha)+ 2*sin(\alpha)*cos(\alpha)=0\\cos^2(\alpha)-sin^2(\alpha)-2*sin^2(\alpha)+ 2*sin(\alpha)=0\\1-sin^2(\alpha)-sin^2(\alpha)-2*sin^2(\alpha)+2*sin(\alpha)=0\\-4*sin^2(\alpha)+2*sin(\alpha)+1=0[/tex]
we can use the quadratic formula to find what sine of alpha is:
[tex]sin(\alpha)=\frac{-2+/-\sqrt{4-4(-4)(1)} }{2(-4)} =\frac{-1+/-\sqrt{5} }{-4}[/tex]
and for [tex]sin(\alpha)[/tex] to be positive (acute angle in the first quadrant) the answer is:
[tex]sin(\alpha)=\frac{1+\sqrt{5} }{4} \\\alpha= arcsin(\frac{1+\sqrt{5} }{4})\\\alpha= 54^o[/tex]
