Question

given the right triangle shown below with one non - right angle of 42° and an adjacent side of length 15: the measure of the other non - right angle is and the lengths of the other sides are: b ≈ c ≈ round your answers to one decimal place. be sure to include the degree symbol in your answer for any angle measured in degrees. question help: video submit question

Explanation:

Step1: Find the other non - right angle

In a right triangle, the sum of the non - right angles is \(90^{\circ}\). Let the other non - right angle be \(x\). We know one non - right angle is \(42^{\circ}\), so \(x + 42^{\circ}=90^{\circ}\). Solving for \(x\), we get \(x = 90^{\circ}- 42^{\circ}=48^{\circ}\).

Step2: Find the length of side \(b\) (opposite the \(42^{\circ}\) angle)

We can use the tangent function. The tangent of an angle \(\theta\) in a right triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\theta = 42^{\circ}\), the adjacent side is \(15\) and the opposite side is \(b\). So \(\tan(42^{\circ})=\frac{b}{15}\). Then \(b = 15\times\tan(42^{\circ})\). Calculating \(\tan(42^{\circ})\approx0.9004\), so \(b\approx15\times0.9004 = 13.506\approx13.5\) (rounded to one decimal place).

Step3: Find the length of side \(c\) (the hypotenuse)

We can use the cosine function. The cosine of an angle \(\theta\) in a right triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\theta = 42^{\circ}\), the adjacent side is \(15\) and the hypotenuse is \(c\). So \(\cos(42^{\circ})=\frac{15}{c}\). Then \(c=\frac{15}{\cos(42^{\circ})}\). Calculating \(\cos(42^{\circ})\approx0.7431\), so \(c\approx\frac{15}{0.7431}\approx20.2\) (rounded to one decimal place).

Answer:

The measure of the other non - right angle is \(48^{\circ}\), \(b\approx13.5\), \(c\approx20.2\)