CPM Homework Help : CCG Problem 5-46

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 $\tan23^\circ\approx0.424$ .

A right triangle with a base of 18 and a hypotenuse of x. An angle opposite the 18 side is 67 degrees.

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 $180^\circ$ .

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

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.

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

The third angle is 23 degrees.

The short side of the triangle is y.

Hint (a):

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

Answer (a):

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

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):

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