Question

solve the right triangle.
write your answers as integers or as decimals rounded to the nearest tenth.
cd =
de =
m∠d = °

Explanation:

Step1: Find \(m\angle D\)

In a right - triangle, the sum of the two non - right angles is \(90^{\circ}\).
If \(\angle C = 59^{\circ}\) and \(\angle E=90^{\circ}\), then \(m\angle D=90^{\circ}-\angle C\).
\(m\angle D = 90^{\circ}- 59^{\circ}=31^{\circ}\)

Step2: Find \(CD\) (hypotenuse)

We know that \(\cos C=\frac{CE}{CD}\), where \(CE = 4\) and \(\angle C = 59^{\circ}\).
By the formula \(CD=\frac{CE}{\cos C}\), substituting \(CE = 4\) and \(\cos59^{\circ}\approx0.515\)
\(CD=\frac{4}{\cos59^{\circ}}\approx\frac{4}{0.515}\approx7.8\)

Step3: Find \(DE\) (opposite side to \(\angle C\))

We know that \(\tan C=\frac{DE}{CE}\), where \(CE = 4\) and \(\angle C = 59^{\circ}\), \(\tan59^{\circ}\approx1.664\)
By the formula \(DE = CE\times\tan C\)
\(DE=4\times\tan59^{\circ}\approx4\times1.664 = 6.7\)

Answer:

\(m\angle D = 31^{\circ}\), \(DE\approx6.7\), \(CD\approx7.8\)