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 3 and 4, angle θ at Q, for trigonometric function exercises.

Accepted Answer

Hello, everyone. We have been asked to determine the missing side in the given triangle using the Pythagorean theorem. Also, we want to write the value of the six trigonometric ratios. We have a right triangle with the right angle at R and we have theta at angle Q and Pr we know is three and QR is four. So we are gonna look first for the value of P Q recall that the Pythagorean theorem says that the sum of the square of the legs of a right triangle is equal to the square of the hypotenuse were often known as A squared plus B squared equals C squared. I'm going to call three A and four B and X is gonna be my missing side. So I will have three squared plus four squared equals X squared, nine plus 16 equals X squared, 25 equals X squared, taking the square root of both sides X will equal five. So this means side P Q is five, which is part of our solution. That's part of our answer. We were looking for the missing side. We also know side pr is three inside QR is four. So now we're gonna work on our trig ratios. I'm gonna start with sign. So the sign of theater is gonna be the opposite over the hypotenuse. So the opposite side over the hypotenuse. So the site opposite theta is pr over the hypotenuse, which is P Q. So this gives us 3/5 as our sign ratio. Next cosine of theta, which is the adjacent side over the hypotenuse. So adjacent to theta is side QR and that's over our hypo hypotenuse, excuse me, which is P Q. So QR was four, P Q was five. So the cosine of theta is 4/5. Next, we have the tangent of theta which is the opposite side over the adjacent side. So opposite of theta is pr we're gonna divide that by the adjacent side which is QR and we will have three divided by four. So 3/4 we have half of our ratios done three more to go. Next, I'm gonna go with C can of theta. So co C can is the same as one divided by the sign of theta, which also is the same as saying we are going to invert the sign of the. So 3/5 inverted would be five thirds next C can of theta is the same as one divided by the cosine of theta. So again, we will invert the cosine, we have the fraction 4/5 inverted. That would be 5/4. And finally, we have the cotangent of theta which is one divided by the tangent of theta. So we can invert the tangent of the, which will give us four thirds. So our six ratios, the sine of theta is 3/5 cosine of theta is 4/ tangent of theta is 3/4 co C of theta is five thirds C can of theta is 5/4 and the code of theta is four thirds. That is all of our ratios and our missing side 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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