How do I find the other two angles when only 90 degrees is shown?

Question

This is from a question from my homework:

"Use the relationship between 6 in. and 9 in. sides to find the unknown angle."

The unknown angle is the little circle thing.

http://i25.photobucket.com/albums/c68/slt212/triangle.jpg

Can you help me please?

Answer 1

The little circle thing θ is called "theta."

Since the triangle is a right triangle (one angle = 90 degrees), you can use the following formulas:

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent

The lengths of all three sides of the triangle are given, so you can use any of the above formulas.

In relation to where θ is, the opposite side is the one that is 6 inches.  The adjacent side is the one that is 9 inches.  The hypotenuse is the longest side, 10.8 inches.

sin θ = 6/10.8.  Take the arcsine (or inverse sine) of both sides to get θ by itself:

θ = arcsin(6/10.8)

θ = 33.75 degrees

(or about 34 degrees)

The sum of a triangle's angles equals 180 degrees.

So, to find the other angle, subtract 90 and 33.75 degrees from 180:

180 - 90 - 33.75 = 56.25 degrees. (or about 56 degrees)

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As I stated before, since you were given the lengths of all three sides, you could also have used the formulas for cos θ and for tan θ to solve the problem:

Using the formula for cos θ:

cos θ = adjacent / hypotenuse

cos θ = 9/10.8.  Take the arccosine (or inverse cosine) of both sides:

θ = arccos(9/10.8)

θ = 34 degrees

180 - 90 - 34 = 56 degrees

Using the formula for tan θ:

tan θ = opposite / adjacent

tan θ = 6/9.  Take arctan of both sides:

θ = arctan(6/9)

θ = 34 degrees

180 - 90 - 34 = 56 degrees.

Hope that helped!