91Ó°ÊÓ

In Problems 81–92, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places.

$4\cos \left(3x\right)-e^x=1,x>0$

What are the zeros of$f\left(x\right)=4{\sin }^{2}x-3$on the interval$[0,2\mathrm{Ï€}]$?

Establish each identity.

${(\sin \mathrm{Î\pm }+\cos \mathrm{β})}^2+(\cos \mathrm{β}+\sin \mathrm{Î\pm })(\cos \mathrm{β}-\sin \mathrm{Î\pm })=2\cos \mathrm{β}(\sin \mathrm{Î\pm }+\cos \mathrm{β})$

In Problems 91-93, find the real zeros of each trigonometric function on the interval$0≤\mathrm{θ}<2\mathrm{π}$

solve each equation on the interval

$0≤\mathrm{θ}<2\mathrm{π}$

$\sin \mathrm{θ}+\cos \mathrm{θ}=\sqrt{2}$

Establish each identity.

${(\sin \mathrm{Î\pm }-\cos \mathrm{β})}^2+(\cos \mathrm{β}+\sin \mathrm{Î\pm })(\cos \mathrm{β}-\sin \mathrm{Î\pm })=2\cos \mathrm{Î\pm }(\sin \mathrm{Î\pm }-\cos \mathrm{β})$

solve each equation on the interval

$0≤\mathrm{θ}≤2\mathrm{π}$

$\sin \mathrm{θ}-\cos \mathrm{θ}=-\sqrt{2}$

Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets $12$inches wide. After marking off a length of $4$inches from each edge, this length is bent up at an angle $\mathrm{θ}$. See the illustration. The area $A$of the opening as a function of $\mathrm{θ}$is given by

$A\left(\mathrm{θ}\right)=16\sin \mathrm{θ}(\cos \mathrm{θ}+1)0°<\mathrm{θ}<90°$

(a) In calculus, you will be asked to find the angle $\mathrm{θ}$that maximizes $A$by solving the equation

$\cos \left(2\mathrm{θ}\right)+\cos \mathrm{θ}=0,0°<\mathrm{θ}<90°$

Solve this equation for $\mathrm{θ}$

(b)What is the maximum area $A$of the opening?

(c)Graph $A=A\left(\mathrm{θ}\right),0°≤\mathrm{θ}≤90°$, and find the angle $\mathrm{θ}$that maximizes the area $A$. Also find the maximum area. Compare the results to the answer found earlier.

What are the zeros of$f\left(x\right)=2\cos \left(3x\right)+1$ on the interval $[0,2\mathrm{Ï€}]$?

solve each equation on the interval

$0≤\mathrm{θ}<2\mathrm{π}$

$\tan \left(\mathrm{θ}\right)+\sqrt{3}=\frac{1}{\cos \left(\mathrm{θ}\right)}$