How to find missing angles using the 30-60-90 special right triangle.

Question

if opposite 60° (b) is 5_{√2, find what are the missing lengths. what is the length opposite 30°(a), and the length opposite 90°} (c).

Answer 1

the following assume you mean one side length is 5sqr2. ( It's written a little ambiguously, as if it might have meant the 5th root of 2)

a 30-60-90 triangle has corresponding sides in a ratio of 1-sqr3-2

if the side opposite 60 degrees is 5sqr2 which is about 7.1

divide by sqr3 to get the side opposite 30: (5sq2)/sqr3 = (5/3)sqr6 = about 4.1

multiply by 2 to get the side opposite 90: (10/3)/sqr6 = the hypotenuse =about 8.2

the sides are (5/3)sqr6 - 5sqr2 - (10/3)sqr6 corresponding to sides opposite 30-60-90 degrees

check the answer with the Pythagorean Theorem

hypotenuse squared = adjacent side squared + opposite side squared

(10/3)sqr6 squared = sum of the other two sides squared

(100/9)6 = (25/9)6 + 25(2)

200/3 = 50/3 + 50

200/3 = 50/3 + 150/3

200/3 = 200/3

if you didn't know the 1 - sqr3 - 2 ratio for a 30-60-90 triangle you could have used the law of sines or cosines to get the lengths of the remaining two sides

the law of cosines reduces to the Pythagorean Theorem when the angle is 90 degrees

but you need to know 2 sides to use the law of cosines

law of sines is sinA/a = sinB/b = sinC/c

or a/sinA = b/sinB = c/sinC

for B=60, b=5sqr2 c/sin90 = (5sq2)/sin60

c/1 = (5sqr2)/(sqr3/2) = (5sqr2)(2/sqr3)=(10sqr2)/sqr3

c= (10/3)sqr6=hypotenuse = side opposite 90 degrees

if you really meant the side oppose 60 was the 5th root of 2 = 2^(1/5)

then divide by sqr3 to get the side opposite 30: ((2)^1/5)/sqr3

then multiply by 2 to get the side opposite 90: (2^(6/5))/sqr3