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

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 here we are asked to determine the missing side in the given triangle using the Pythagorean theorem. And we also need to write the value of these six trigonometric ratios for angle P. Here we are given a right triangle with angles QP and R where angle Q is given as 90 degrees, we then have side links P and R and the hypotenuse Q, we are also given P is equal to nine and R is equal to 40. So to begin this problem, we can start by finding the value of the missing side. And here looking at the given information, we can see that the missing side is the hypotenuse Q. And so we need to recall the Python theorem which states that Q squared is equal to R squared plus P squared. So now we just want to substitute in our given values. And so we have Q squared is equal to squared plus nine squared. And now just solving for Q, we need to take the square root of both sides. And so we have Q is equal to the square root of 40 squared plus nine squared. And now just simplifying our expression, we find that the value for Q is 41. And so now we have already determined the missing side. So we can move on to the next part of the problem where we need to find the ratios for each of the trigonometric functions for angle P. So we can begin here by finding the ratio for sine of P. And so we just need to recall that sign is opposite over hypotenuse. So in reference to our given triangle, this is P lowercase P divided by Q and now just substituting in our values that we have, we have nine divided by 41 as the ratio for a sign of P. And now we can move on to cosine of P recalling that cosine is adjacent over hypotenuse. And so we have R divided by Q and substituting in our values we have 40 divided by 41. Now we can move on to tangent of P and we just need to recall that tangent is opposite over adjacent. So here we have P lowercase P divided by R and now substituting in our values, we have nine divided by 40. Now, we can move on to the next function where we will have co sequent of P. And we just need to recall that coin is equivalent to one divided by sine. So here cos of P is equivalent to one divided by sine of P. And now we just want to substitute in the ratio we found for a sign of P. And so we have one divided by nine divided by 41. And now just simplifying we have 41 divided by nine. Next, we can move on to sin of P and sin of P. We just need to recall is equivalent to one divided by cosine of P. So now just substituting in the ratio we found for cosine, we have one divided by 40 divided by 41. Simplifying we have the ratio 41 divided by 40. Lastly, we can move on to finding cotangent of P and we just need to recall that cotangent of P is the same as one divided by tangent of P substituting in the ratio. We found we have one divided by nine divided by 40. And simplifying we have 40 divided by nine. And with this, we have found our six trigonometric ratios as well as the missing side in our given triangle. And so we are done with both steps of the problem. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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

Key Concepts

Pythagorean Theorem

Trigonometric Functions

Rationalizing Denominators

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