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

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 6 and 10, angle θ at Q, for trigonometric function exercises.

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 have a right triangle where the right angle is at angle R, the triangle is Pr Q and Q is also angle theta we are told that the hypotenuse which is side P Q has a value of 10. And then QR has a value of six. We need to find the value of the missing side, which is pr since we were asked to use the Pythagorean theorem, recall that the Pythagorean theorem states that the sum of the squares of the legs of a right triangle are equal to the square of the hypotenuse, commonly referred to as A squared plus B squared equals C squared C must be our hypotenuse. And since our hypotenuse is across from the right angle, in this case, our hypotenuse is P Q. So 10 is gonna be our C value. Pr is my unknown. I'm gonna call that A and QR which has a value of six is gonna be B. So using the Pythagorean theorem, I get a squared plus six squared equals squared. Doing out my squares, I still have a squared. Now plus 36 equals 100. I wanna get the A squared term by itself. So I'm subtracting 36 from both sides and I have a squared equals 64. The opposite of a square is a square root. So square rooting both sides, I get that A equals eight. So this means that side pr has a value of eight. So we've done the first part, we found our missing side. We're now gonna use this to find our trig ratios. I'm gonna start with sign. So sign of data sign is the opposite. So side pr divided by the hypotenuse which is side P Q. So Pr has a value of eight P Q has a value of 10, 8/10 can be reduced to 4/5. So that is our value of signed. The next, the cosine of theta cosine is gonna be the adjacent side divided by the hip hot. So the adjacent side here is QR divided by the hippo which is P Q or Q P and this will be six for QR divided by 10, for the hypotenuse 6/10 can be reduced to 3/5. And that will be our cosign of theta. Next, we find the tangent of the tangent is the opposite of theta. So side pr divided by the adjacent side which is QR. So Pr has a value of eight. QR has a value of six, 8/6 can be reduced to four thirds. And that is our value for the tangent of data. Next can of theta is the same as one divided by the sign of the. So we are going to invert the sign and that will give us the CO C can. So this would be P Q divided by pr which P Q is 10, pr is eight. So 8/10 can be reduced to 5/4. And that is gonna be our C can of the next, the secant of theta is the same as one divided by the cosine of the. So we can invert the cosine and that will give us our sea can. So that would be P Q divided by QR P Q has a value of 10. QR has a value of six. So 6/10 can be reduced to five thirds which is our C can of theta. And our last one would be the cotangent of the, which is the same as one divided by the tangent of the, which can be again looked at as the inverse. So flipping it over. So this would be QR divided by Pr QR has a value of six. Pr has a value of eight and 6/8 can be reduced to 3/4. So 3/4 is our cotangent of the. So here are all six trig ratios. Sine is 4/5 cosine of theta is 3/ tangent of theta is four thirds co C can of theta is 5/4 C can of theta is five thirds cotangent of theta is 3/4. And our missing side pr had a value of eight. 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

Trigonometric Functions

Right Triangle Properties

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