45-45-90 Triangle Problem

Question

A right triangle contains a 45 degree angle, 90 degree, and an unknown degree, y. The hypotenuse equals 50, while the side opposite of the 45 degree angle is x. Use a trigonometric function to find the value of y and x.

Answer 1

A 45-45-90 triangle is an isosceles triangle.  This is because 2 of the angles are the same and the length of the legs are the same.  90 degrees is the angle opposite of the hypotenuse 50.

 

We can use sine function.

 

sin(45) = x / 50

 

50sin(45) = x

 

50(√(2) / 2) = x

 

25√(2) = x

 

 

 

To find y, we can use the cosine function.

 

cos(y) = x / 50

 

 

Use the value of x that we just found into this equation.

 

cos(y) = (25√(2)) / 50

 

cos(y) = √(2) / 2

 

y = cos^-1(√(2) / 2)

 

y = 45

 

Answer 2

Since we know that the three angles of a triangle must add to a total of 180 degrees

and we have a 90 degree and a 45 degree angle then we have an isosceles triangle

and the angles are 45, 45, 90 but...

 

 

To find the length of the side x we use the sine function that sin y = opposite/hypotenuse

sin45 = x/50

50(sin45) = x

x = 50(√2/2)

x = 25√2

 

Note:  Since the problem stated that we need to use a trigonometric function to find x and y

then instead of going the simple route to find y... let's use the value we found for x and the

cosine function to determine y...

 

cos(y) = adj/hyp = (25√2)/50 = √2/2

y = cos^-1(√2/2)

y = 45 degrees