How do I find the cosine of a triangle with an angle of 30 and a hypotenuse of 15

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Triangles don't have cosines.  Angles do. To find the cosine of an angle in a RIGHT triangle, divide          (the length of the leg adjacent to the angle) by          (the length of the hypotenuse). If the angle is 30°, then the cosine is  1/2 of √3  (about 0.866...) -- The cosine of 30° is always 0.866... , but if the triangle is not a RIGHT triangle, then (adjacent side)/(longest side) is not the cosine of the angle. -- The size of the triangle doesn't matter.  It can be a picture drawn on the head of a pin, or the size of the solar system. If it's a RIGHT triangle and it has a 30° angle in it, then (adjacent leg)/(hypotenuse) is 0.866... .  It's the ratio that counts. ___________________________________ cosine(30°) = 1/2 of √3 . If the hypotenuse of the right triangle is 15 and it has a 30° angle in it, then                         (leg adjacent to the 30° angle) / 15 = 1/2 of √3 . Multiply each side by 15:                           (leg adjacent to the 30° angle)        = 7.5 √3                            Leg adjacent to the 30° angle       =  12.99 . This gives you a hint of one way that cosines (also sines and tangents) can be useful.  If you need to know the length of something outside, but there's no way to measure it ... the distance from you to a tree on the other side of the river, the length of one side of a piece of legal property, the height of a big tree or the wall of a tall building ... you can often lay out a right triangle on the ground, connected to the unknown length or distance.  Then you measure one of the OTHER sides of the triangle, and one of the ANGLES, and by using the cosine (or sine or tangent) of the angle, you can calculate the length or the distance that you can't measure.   

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