Estelle is trying to find $x$ in the triangle at right. She lost her scientific calculator, but luckily her teacher told her that $\sin23^\circ\approx0.391$, $\cos23^\circ\approx0.921$, and $\tan23^\circ\approx0.424$.  

1. Write an equation that Estelle could use to solve for $x$.

   Step 1(a):

   Use the Triangle Angle Sum Theorem to calculate the missing angle, this means all the angles in a triangle total $180^\circ$.  Label added, third angle, theta = 23 degrees.

   $67^\circ + 90^\circ + θ = 180^\circ$
   $-157-157$
   $θ = 23^\circ$

   Step 2(a):

   Place a "$y$" on the missing side. Write the trigonometric ratios for the angle, reference the Math Note box in Lesson 5.1.2 for extra help. Label added, short side is y.

   $\sin23^\circ=\frac{y}{x}\ \ \ \ \cos23^\circ=\frac{18}{x}\ \ \ \tan23^\circ=\frac{y}{18}$

   

   

   Hint (a):

   Which equation can be used to solve for $x$?

   Answer (a):

   ${\cos 23^\circ=\frac{18}{x}}$

2. Without a calculator, how could Estelle find $\sin67^\circ$? Explain.

   Step 1(b):

   $\text{sin}(67^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{18}{x}$

   Step 2(b):

   $\text{sin}(67^\circ)=\frac{18}{x}=\text{cos}(23^\circ)$

   Step 3(b):

   $\cos23^\circ\approx0.921$

   Answer (b):

   $\sin67^\circ=\cos23^\circ\approx0.921$