Solving for a side in right triangles with trigonometry

we're asked to solve the right triangle shown below give the lengths to the nearest tenth so when they say solve the right triangle we can assume that they're saying hey figure out the lengths of all the sides so whatever a is equal to whatever B is equal to and also what are all the angles of the right triangle we get they given two of them we might have to figure out this third right over here so there's multiple ways to tackle this but we'll just go and we'll just try to tackle side XW first try to figure out what a is and I'll give you a hint you can use a calculator and using a calculator you can use your trigonometric functions that we've looked at a good bit now so I'll give you a few seconds to think about how to figure out what a is well what do we know we know this angle Y right over here we know the side adjacent to angle Y and length a this is the side that's the length of the side that is opposite that is opposite to angle Y so what trigonometric ratio what trigonometric ratio deals with the opposite and the adjacent so if we're looking at angle Y relative to angle Y this is the opposite and this right over here is the adjacent well if we don't remember we can go back to sohcahtoa so Toa sign deals with opposite and hypotenuse cosine deals with adjacent and hypotenuse tangent deals with opposite over adjacent opposite over adjacent so we can say that the tangent the tangent of 65 degrees of that angle of 65 degrees is equal to the opposite the length of the opposite side which we know has length a over the length of the adjacent side which they gave us in the diagram which has length which has length 5 and I said well how do I figure out a well we can use our calculator to evaluate what the tangent of 65 degrees are and then we can solve for a and actually if we just want to get the expression explicitly solving for a we can just multiply both sides of this equation times 5 so let's do that 5 x times 5 the cancel out and we are left with if we flip the the equal around we're left with a is equal to five times the tangent of 65 of 65 degrees so now we can get our calculator out and figure out what this is to the nearest tenth my handy ti-85 out and I have five times that not the the tangent not I didn't need to press that second right over there just a regular tangent of 65 degrees and I am like I get if I round to the nearest tenth like they asked me to I get ten point seven so this is so a is approximately equal to ten point seven I say approximately because I rounded it I rounded it down I didn't this is not the exact number but a is equal to ten point seven so we now know that this has length ten point seven approximately there's several ways that we can try to tackle B and I'll let you pick the way you want to but then I'll just do it the way I would like to so my next question to you is what is the length of the side why W or what is the value of B well there's several ways to do it this is the hypotenuse so we could use trigonometric functions that deal with adjacent over hypotenuse or opposite over hypotenuse or we could just use the Pythagorean theorem we know two sides of a right triangle we can come up with the third side I won't go with I will go with using trigonometric ratios since that's what we've been working on a good bit so this is length the B that's the length of the hypotenuse so this side WI is the hypotenuse and so what trigonometric ratio so we can decide what we want to use we could use opposite in hypotenuse we could use adjacent and hypotenuse since we know that XY is exactly five and we don't have to deal with this approximation let's use that side so what trigonometric ratios deal with adjacent and hypotenuse well we see from sohcahtoa cosine deals with adjacent over hypotenuse so we could say that the cosine of 65 degrees cosine of 65 degrees is equal to the length of the adjacent side which is five over the length of the hypotenuse which has length B and then we can try to solve for B you multiply both sides times B you're left with B times cosine of 65 degrees is equal to five and then to solve for B you could divide both sides by cosine of 65 degrees this is just a number here so we're just dividing we have to figure it out with our calculator but this is just going to evaluate to some number so we can divide both sides by that by cosine of 65 degrees cosine of 65 degrees and we're left with B is equal to five over the cosine of 65 degrees so let us now use our calculator to figure out the length of B length of B is five divided by cosine of 65 degrees and I get if I round to the nearest tenth 11 point eight so B is approximately equal to rounded to the nearest tenth eleven point eight so B is equal to eleven point eight and then we're almost done solving this right triangle and you could have figured this out using the Pythagorean theorem as well saying that five squared plus ten point seven squared should be equal to B squared and hopefully you would get the exact same answer and the last thing we have to figure out is the measure of angle W angle W right over here so I'll give you a few seconds to think about what the measure of angle W is well here we just have to remember that the sum of the angles of a triangle add up to 180 degrees so angle W plus 65 degrees that's this angle right up here plus the right angle this is a right triangle they're going to add up to 180 degrees so all we need to do is well we could simplify the left-hand side right over here 65 plus 90 is 100 fifty-five so angle w+ 155 degrees is equal to 180 degrees and then we get angle w if we subtract 155 from both sides angle w is equal to 25 degrees and we are done solving the right triangle shown below