Consider the triangle below

First we must understand the definition of complementary angles.  Complementary angles are a pair of angles that add up to 90 degrees.  Looking at the triangle above, we see that and  add up to 90, so they are complementary.

Recall that the definition of sine is  and cosine is .  

Next, list out the sine and cosine of  and .

Notice that 

And 

We have just shown that the sine of the two angles is equal to the cosines of their complements.  This is true for any acute angles.

You know that .  We use this information to set up our work in the following way:

To solve for , we need to use the definition of complementary.  Both of these angles should add up to be 90, so we can set up this problem by setting the sum of the two angles equal to 90.

This relationship is true for any pair of angles that are complementary (i.e. any two angles that add up to 90). We can see this in obtuse triangles that have two acute angles that add up to 90, we can see it in acute triangles, and equilateral triangles as well.  Beyond triangles this relationship is also true.

We will use the definition of complementary to solve for .   and  must sum up to be 90 by the definition of complementary angles.  We are able to set up the following equation:

 (combine like terms)

Using our relationship we know that .  We can set up the following equation to solve for 

We can double check our work using the definition of complementary angles.  That means that .  We know that our answer must be correct because .

First, we need to think about what it means to be a square.  Squares have 4 congruent sides and 4 congruent angles that must add up to 360.  So each angle must be 90-degree angles.  The diagonal of the square cut two of these 90-degree angles in symmetrical halves to form two right triangles within the square.  Now we have two identical right triangles with legs of length 5 and angles of 90, 45, 45.

Recall what we know about our sine and cosine relationships:

So we can use either sine or cosine to solve for the hypotenuse.  We will first use cosine:

We can also use sine since their relationship makes them equivalent

And now we can confirm using the Pythagorean Theorem to check our work:

Recall our sine and cosine relationship of complementary angles.  The sine of any acute angle is equal to the cosine of its complement:

We know that  and . 

We can use this information to set up the following equation to solve for .

 ()

 (multiply both sides by )

We begin by noting that this triangle is an equilateral triangle as stated in the problem.  This means that all sides are congruent and all angles are congruent.  Since a triangle’s angles must add up to 180 degrees, we can deduce that each of the three angles is 60 degrees.  By drawing a vertical auxiliary line down from vertex A perpendicular to the base of the triangle we are evenly dividing the equilateral triangle into two right triangles.  We can update the triangle as such:

We also know that each of the three angles of the equilateral triangle are equal to 60 degrees.  Since the auxiliary line splits the equilateral evenly into two right triangles, we know that  and that .  Again, we can update our triangle:

Now we can begin to use our sine and cosine relationships of complementary angles to solve for .  Recall the following:

Since  we can use either one to solve for this problem.

or

We can deduce that  and are complementary angles because triangle  is a right triangle.  We have enough information from this figure to find .  Once we have this measurement we will be able to use the sine cosine relationship of complementary angles to solve for .

Since  is a straight line segment, .  Plugging in our values we can find

We could just use our knowledge that a triangle must add up to 180 degrees to solve for  but that is not what the question is asking.  We must now use the fact that ).  We can set up the following equation to solve for  .