Question

find m∠c.
write your answer as an integer or as a decimal rounded to the nearest tenth.
m∠c = °

Explanation:

Step1: Identify triangle type and trigonometric ratio

This is a right - triangle \( \triangle ABC \) with \( \angle A = 90^{\circ} \), \( AC=\sqrt{11} \), and \( AB = \sqrt{13} \). To find \( m\angle C \), we can use the tangent function. The tangent of an angle in a right - triangle is defined as the ratio of the opposite side to the adjacent side. For \( \angle C \), the opposite side is \( AB=\sqrt{13} \) and the adjacent side is \( AC = \sqrt{11} \). So, \( \tan(C)=\frac{AB}{AC}=\frac{\sqrt{13}}{\sqrt{11}} \)

Step2: Calculate the value of the tangent ratio

First, calculate \( \frac{\sqrt{13}}{\sqrt{11}} \). We know that \( \sqrt{13}\approx3.606 \) and \( \sqrt{11}\approx3.317 \). Then \( \frac{\sqrt{13}}{\sqrt{11}}\approx\frac{3.606}{3.317}\approx1.087 \)

Step3: Find the angle whose tangent is approximately 1.087

We use the inverse - tangent function (arctangent). \( \angle C=\arctan(1.087) \). Using a calculator, \( \arctan(1.087)\approx47.3^{\circ} \) (rounded to the nearest tenth)

Answer:

\( 47.3 \)