Find exact values or expressions for sin A, cos A, and tan A. ...

Question

Find exact values or expressions for sin A, cos A, and tan A. See Example 1.

Right triangle with sides 20, 21, and hypotenuse 29, showing angle A and a right angle.

Accepted Answer

Hey, everyone in this problem, we're given a triangle and we're asked to determine the exact value of sine of alpha cosine of alpha and tangent of alph the triangle we're given is a 90 degree triangle. OK? A right angle triangle, the hypotenuse has a length of 61. We have angle alpha with the opposite side of 11 and an adjacent side of 60. We're given four answer choices A through D, each of them containing different exact values for our three trick ratios that we're interested in. And we're gonna come back to those as we work through this program. So let's go ahead and start with sine of alpha and recall that sign of an angle theta is going to be equal to the opposite side divided by the hyper. So in this case, we're interested in angle alpha. So the opposite side to angle alpha, OK, we go across to the side that does not touch the corner where that angle is, that's going to be 11 in the hypo. The longest side length across from a right angle is 61. And so side of alpha is going to be 11 divided by looking at our answer choices, we can already eliminate two of them. Option A and option C both have the correct value for sign of alpha. But option B and D do not, they have 60 divided by 61 and 61 divided by 11 respectively, which are not the correct answers. So we can eliminate those. Now, let's move to cosign and recall that cosine of an angle alpha can be written as the adjacent side divided by the hypos. OK. So in this case, we're looking at angle alpha cosine of alpha is going to be equal to the adjacent side, which is 60 divided by the hypotenuse, which is 61. And so coastline of alpha is 60 divided by 61. So both answer choice A and C both have this value for cos. So we're still sticking with either of those answer choices could be correct. Now, let's move to the tangent and recall that tangent of data, OK? Is going to be equal to the opposite side divided by the adjacent side. Now, we could do tangent of data a little bit different. We could also recall that tangent of data is going to be sine data divided by cosine data. And we have sine beta and we have cosine data. So we could use that relationship to find this as well. We're just gonna stick with this basic definition of the opposite side divided by the adjacent side and it's gonna be a little bit simpler. We don't have to worry about dividing anything. So we get T of alpha is going to be equal to the opposite side divided by the adjacent side 60. And so looking at our answer choices, we can see that answer choice A has the correct value for tangent as well. And so this is gonna be our correct answer option. C had the reciprocal of what we found to be correct for tan alpha. So we found that sign of alpha is equal to 11 divided by 61 cosine of alpha is equal to 60 divided by 61 and tangent of alpha is equal to divided by 60 corresponding with option A. Thanks everyone for watching. I hope this video helped see you in the next one.

Find exact values or expressions for sin A, cos A, and tan A. See Example 1.

Key Concepts

Definition of Sine, Cosine, and Tangent

Using Reference Triangles and Special Angles

Trigonometric Identities and Relationships

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