Question

1. find the exact value of the missing side.
2. convert 144° to radians.
3. convert \\(\frac{14\pi}{9}\\) to degrees.

Explanation:

Problem 1:

Step1: Identify the triangle type

It's a right - triangle, so we can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs. Let the missing side be \(a\), \(b = 16\), \(c=24\).

Step2: Apply the Pythagorean theorem

We have \(a^{2}+16^{2}=24^{2}\). So \(a^{2}=24^{2}-16^{2}\). Using the difference of squares formula \(x^{2}-y^{2}=(x + y)(x - y)\), we get \(a^{2}=(24 + 16)(24-16)=(40)(8)=320\). Then \(a=\sqrt{320}=\sqrt{64\times5}=8\sqrt{5}\).

To convert degrees to radians, we use the formula \(radians=\text{degrees}\times\frac{\pi}{180}\).

Step1: Substitute the value

For \(144^{\circ}\), we have \(144\times\frac{\pi}{180}\).

Step2: Simplify the fraction

Simplify \(\frac{144}{180}=\frac{4}{5}\). So the value in radians is \(\frac{4\pi}{5}\).

To convert radians to degrees, we use the formula \(\text{degrees}=\text{radians}\times\frac{180}{\pi}\).

Step1: Substitute the value

For \(\frac{14\pi}{9}\), we have \(\frac{14\pi}{9}\times\frac{180}{\pi}\).

Step2: Simplify the expression

The \(\pi\) terms cancel out, and \(\frac{14\times180}{9}=14\times20 = 280\).

Answer:

\(8\sqrt{5}\)

Problem 2: