Question

find the missing side. 23 30° y y = ?

Explanation:

Step1: Identify the trigonometric ratio

We have a right - triangle with hypotenuse \(c = 23\), angle \(\theta=30^{\circ}\), and the adjacent side to the angle \(30^{\circ}\) is \(y\). The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos(30^{\circ})=\frac{y}{23}\).

Step2: Recall the value of \(\cos(30^{\circ})\)

We know that \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\approx0.866\).

Step3: Solve for \(y\)

From \(\cos(30^{\circ})=\frac{y}{23}\), we can re - arrange the formula to solve for \(y\). Multiply both sides of the equation by \(23\):
\(y = 23\times\cos(30^{\circ})\)
Substitute \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\) into the equation:
\(y=23\times\frac{\sqrt{3}}{2}=\frac{23\sqrt{3}}{2}\approx\frac{23\times1.732}{2}=\frac{39.836}{2} = 19.918\approx20\) (or we can keep it in exact form \(\frac{23\sqrt{3}}{2}\))

Answer:

\(y=\frac{23\sqrt{3}}{2}\) (or approximately \(19.92\))