In Exercises 1–12, solve the right triangle shown in the figure. ...

Question

In Exercises 1–12, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. a = 30.4, c = 50.2

Right triangle labeled with sides p, q, r and angles Q, R, P for solving exercises in trigonometry.

Accepted Answer

Hello, everyone. We are asked to find the solution for the right angle. In the given figure, we will be expressing angles to the nearest 10th of a degree and rounding lengths to two decimal places. We have a right triangle, QR P with the right angle at angle R, our sides are labeled in lower case letters P Q. And our hypotenuse is R, we are told that P side P is 15.7 and side R is 29.3, we are given four answer choices with varying values for angles P and Q and side Q. The first thing I would like to do is label what we are given. We are told side P is 15.7 and that side R is 29.3. This helps me see what I'm working with and what I have and don't have, I'm gonna start by trying to find the angle at P. So looking at the values we're given side P is opposite angle P and ours are hippo. So thinking of the ratio where we had an opposite of or a hypotenuse that would be sign. So our sign function at angle P is side P. So 15.7 divided by the hypotenuse which is side R 29.3, we are not taking the sign of 15.7 divided by 29.3. Though we need to take the inverse sign. So recall if you're using a calculator, you want to make sure it's in degree mode and we're gonna use the inverse sine function. So sign with the negative one exponent of 15.7 divided by 29.3. And this will give us the measure of the angle at P and we will get 32.4006. So 0.0 or sorry 0. degrees as they want us to round to the nearest 10th for our, for our angles, the angle at P will be 32.4 degrees. So I'm gonna write that on to my diagram, 32.4 degrees is at P next to find the angle at Q. I recall that the sum of the interior angles of a triangle total to 100 80 degrees. So I have 100 80 degrees minus my 90 degree right angle minus 32.4 degrees. And that'll give me the measure of the angle at Q which is 57.6 degrees. So this means that the A Q it's 57.6 degrees. I'm gonna label that in our figure also 57.6 degrees. So we now know all of the angles we have one more side left to find. And we could use the Pythagorean theorem, we could use other trig identities. I think I'm gonna use the Pythagorean theorem. Pythagorean theorem states that the sum of the squares of the legs of a right triangle are equal to the square of its hypo. So also referred to sometimes as A squared plus B squared equals C squared. So I'm gonna say P is my A so 15. squared plus my unknown side which is Q squared equals my hypo squared. So 29.3 squared, I'm gonna do out the squares and then we'll solve for Q, when we do out our squares, we have 246. plus Q squared equals 858.49 to get the Q by itself. I subtract 246.49 from both sides. And I'll leave my Q squared alone on the right hand side, I'm actually gonna bring this right up to the top of the next column. So we have a little more space. So Q squared will equal to find out what Q equals. We do the opposite of a square, which you might recall as a square root. So Q would equal 24.739. And because we want to round our decimals to two places side, Q will be 24.74. Again, I'm gonna put that on my diagram 24.74. And we have all three values. We know angle P is 32.4 degrees. The angle at Q is 57.6 degrees inside Q is 24.74. So now to see which answer choice that matches using side or sorry angle P to start with, it has to be either A or C and then looking for the measure that matches for angle Q, it appears to be A and I'm verifying side Q is 24.74. It is. So our answer choice of A is our correct answer. Have a nice day.

In Exercises 1–12, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. a = 30.4, c = 50.2

Key Concepts

Right Triangle Properties

Trigonometric Ratios

Pythagorean Theorem

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