Question

given the right triangle shown below with one non - right angle of $46^{circ}$ and an adjacent side of length 6: the measure of the other non - right angle is and the lengths of the other sides are: $b\approx$ $c\approx$ 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 interior angles is \(180^{\circ}\), and one angle is \(90^{\circ}\). Let the other non - right angle be \(x\). Then \(x + 46^{\circ}+90^{\circ}=180^{\circ}\). So \(x=180^{\circ}-90^{\circ}-46^{\circ}=44^{\circ}\).

Step2: Find the length of side \(b\) (opposite to \(46^{\circ}\))

We know that in a right triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 46^{\circ}\), adjacent side \( = 6\), and opposite side is \(b\). So \(\tan(46^{\circ})=\frac{b}{6}\), then \(b = 6\times\tan(46^{\circ})\). Calculate \(\tan(46^{\circ})\approx1.0355\), so \(b\approx6\times1.0355 = 6.213\approx6.2\) (rounded to one decimal place).

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

We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 46^{\circ}\), adjacent side \( = 6\), and hypotenuse is \(c\). So \(\cos(46^{\circ})=\frac{6}{c}\), then \(c=\frac{6}{\cos(46^{\circ})}\). Calculate \(\cos(46^{\circ})\approx0.6947\), so \(c\approx\frac{6}{0.6947}\approx8.6\) (rounded to one decimal place).

Answer:

The measure of the other non - right angle is \(44^{\circ}\), \(b\approx6.2\), \(c\approx8.6\)