Suppose ABC is a right triangle with sides of lengths a, b, an...

Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1.

a = 5, b = 12

Accepted Answer

Hey, everyone in this problem, we're asked to determine the missing side in the given triangle using the Pythagorean theorem. And also to write the value of these six trigonometric ratios for angle P. So we're given triangle P QR here we have Q is the right angle. We're told that side P is equal to 12 and side Q is equal to 37. OK. So first thing we want to do is find the missing side. OK. So the missing side in this case is going to be side R. Now we're told to use the Pythagorean theorem, we have this right angle triangle. So we can go ahead and use it and recall that the Pythagorean theorem tells us that the square of the hypos. OK. So Q squared is gonna be equal to the sum of the squares of the other two sides. So Q squared is equal to R squared plus P squared. Now we can substitute in the values we have and solve for our, we have 37 squared is equal to R squared plus 12 squared and we can expand and simplify 1369 is equal to R squared plus subtracting 144 from both sides gives us R squared is equal to 1225. And we can take the square root and we're gonna take the positive route because we're talking about a side length here. So we want that positive value and we get that R is equal to 35. All right. So we have the missing side link. Now, now we're gonna move to part two of this question was which is asking to find a value for the six trigonometric ratios. Now recall that those troop ratios relate the sides of the triangle to the particular angle we're looking at which is going to be angle P, right? So we have all three site links now. So we can go ahead and do exactly what the question is asking. So let's start with sign and recall that sign and angle theta is going to be equal to the opposite side divided by the hypos. So in this case, we have sign of P is equal to now, the side opposite to angle P is side P and the hypos is side Q. So we have P divided by Q and this gives us a sign of P is equal to 12 divided by 30. So, all right. So that's now we can move to the next one and we're gonna move to cosign. So cosine of data, can I recall cosine of data is the adjacent side divided by the hypos so similar to sin but using the adjacent side. So we have cosine of P is gonna be equal to R. OK. That side right next to our angle divided by the hypos Q. And that gives us a cosine of P is going to be equal to divided by 37. All right, we have two down four to go. Let's now work on tangent. And for tangent, we can do this a couple of different ways. OK. We can notice that the tangent is going to be equal to stein of P divided by cosine of P. We have sine, we have cosine. So we can go ahead and use that rule or we can recall that tangent of P is going to be equal to the opposite side divided by the adjacent side. And let me write this story in terms of data first, just a general angle data. So in terms of P, we have that tangent of P is going to be equal to that opposite side, which is P divided by the adjacent side, which is R and we get that 10 of P is equal to 12 divided by 35. And that's the exact same result you would find if you had have done sine of P divided by cosine of P as well. And either method would have worked all right. Three down three to go. We're gonna move to ko again. No, he can't oc of P is going to be equal to one divided by sign of P I recall that it is the reciprocal. So we know sign of P, we can write that coy can of P is going to be one divided by 12, divided by 37 which tells us that CO CNFP is going to be 37 divided by right now. We're gonna do the same for C can. OK. So we have C CFP. OK. This is gonna be the reciprocal of cosine. Can you recall? So seeking it of P is one divided by cosine of P. We know cosine of P. So we get C PM FP is equal to one divided by 35 36. And so CKFP is going to be equal to divided by 35. And for the final trig ratio, we are looking for co tangent, a cotangent of P and cotangent of P recall is equal to one divided by tangent of P. OK. So we have co tangent of P. It's gonna be one divided by 12 35th which tells us that the cotangent of P is gonna be divided by 12. And so once we found that missing sidelines are to be 35 we could write all six of these trig ratios using those sidelines. That's it for this question. Thanks everyone for watching. I hope this video helped

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. a = 6, c = 7

Key Concepts

Pythagorean Theorem

Trigonometric Functions in Right Triangles

Rationalizing Denominators

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