In Exercises 1–8, use the Pythagorean Theorem to find the leng...

Question

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

Right triangle PQR with sides 28 and 53, angle θ at vertex R.

Accepted Answer

Hello, everyone. We are asked to determine the missing side in the given triangle using the Pythagorean theorem. Also, we will write the value of the six trigonometric ratios. We are given a right triangle P. QR with the right angle at angle Q theta is at angle R side P Q measures 28 side pr which is our hypotenuse measures 53. And QR is our unknown side. Recall, the Pythagorean theorem says that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse also sometimes referred to as A squared plus B squared equals C squared. Recall that C must be our hypotenuse. So in this example, C is side pr which is 53 I'm gonna call it 28 A and so side QR is what I'm looking for. And I'm calling that B. So we have 28 squared plus B squared equals 53 squared. 28 squared is 784 plus B squared equals 53 squared, which is subtracting 784 from both sides to get the B squared by itself, we have B squared equals 2025. The opposite of a square is a square root. So to find out what B equals, we take the square root of both sides and the square root of 2025 is 45. So B equals 45 which means side QR is 45. So that's our first part of our answer. We found our missing side. Next, we're gonna use this to find all of our trig ratios. So I'm starting with S sign. So the sign of theta sign is the opposite side over the hypotenuse. So the side, opposite angle, theta is P Q and that's gonna be over or divided by the hypotenuse which is side pr. So P Q is 28 oh divided by Pr which is 53 and that cannot be reduced. So that is our sign of the next cosine of the is the site that's adjacent to theta divided by the hypotenuse. So the leg adjacent to theta is QR divided by the hypotenuse which is pr this will be 45 divided by 53 and that fraction cannot be reduced. So that is our cosine of theta. Next, the tangent of theta is the opposite side from theta divided by the adjacent side. So the opposite side again is side P Q. The adjacent side is QR P Q S value is 28 divided by QR which is that cannot be reduced. So this is the tangent of data. So 28 divided by 45 is the tangent moving on to the of the, which is sometimes referred to as one over the sign of theta. It also could be referred to as the reciprocal of the sign. So this would be the hypotenuse divided by the opposite side. So pr divided by P Q which gives us 53/ and that is our of theta because again, there's nothing that could be reduced there. See can of theta. Again, sometimes we refer to that as one divided by the cosine of theta. It's also the reciprocal of the cosine. So this would be the hypotenuse divided by the adjacent side. So this would be pr divided by QR and that's 53 divided by 45 it cannot be reduced. And that is the sequent of theta. Last one is the cotangent of theta, which sometimes is referred to as one divided by the tangent of theta. Could also be thought about as the reciprocal of theta. I'm sorry, reciprocal of the tangent. So that would be the adjacent side divided by the opposite side. So QR divided by P Q and this would be 45 divided by 28. Again, that cannot be reduced. So this is our cotangent of data. So we have all of our answers. Our missing side. QR was 45. The sign of data is 28 divided by cosine of THETA is 45 divided by tangent of THETA is 28 divided by of theta is 53 divided by C Can of THETA is 53 divided by 45 co tangent of theta is 45 divided by 28. Have a nice day.

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

Key Concepts

Pythagorean Theorem

Right Triangle Trigonometric Functions

Identifying Sides Relative to an Angle

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