The definition of the sine of an angle is .  So here we must determine which side is opposite of angle  and which side is the hypotenuse of this triangle.  We know that side  is the hypotenuse since it is opposite of the right angle.  Side  is directly opposite of angle .  So:

 We are given angle  and we need to find sides  and .  So first we need to think about what relation these sides are to angle .  Side  is the hypotenuse of the triangle since it is opposite of the right angle.  So side must be the side adjacent to angle .  Recall that the trigonometric ratio that corresponds to cosine is .  We can solve for the missing sides by solving for the following equation.

It follows that:

The definition of cosine of an acute angle is .  So here we must determine what sides are adjacent and which is the hypotenuse.  The hypotenuse is the easiest to pick out.  This is the side that is directly across from the right angle in a right triangle.  Our hypotenuse is .  Now we must choose the adjacent side.  Adjacent means next to, and since  is our hypotenuse then  must be our adjacent side.  So:

We are given the adjacent side to angle  and the hypotenuse of the triangle.  We can use this to set up the trigonometric ratio  which we know to be the definition of cosine.  We can solve for angle  using the following equation.

We must begin by manipulating our equation of the side ratios to get fractions that include both of the sides of the same triangles:

 multiply both sides by 

 divide both sides by 

If we look at angle  we see that  is the .  Looking at angle  we see that  is also the .  This is the definition of the tangent of an angle.  So:

We must begin by manipulating our equation of the side ratios to get fractions that include both of the sides of the same triangles:

If we look at angle , we see that  is the .  Looking at angle  we see that  is also the .  This is the definition of the sine function.  So:

There are two approaches to this problem.  We are able to calculate the angle by using the Sine Law.  The Sine Law states:

So we can set  and solve accordingly for angle .

Our other option is to use the area formula we have been but altering it to correspond to angle .  We would draw our vertical line down from the vertex as shown below and our formula would be in the form .

Even though the formula  is using sides and angle , this is a general formula and can be used with any angle in the triangle.  Since we are now working with an obtuse angle rather than an acute angle, we need to do some more work to get the logic right.

Using the figure above, to be able to label the sides we are using correctly, we extend the original triangle horizontally past the obtuse angle and draw a vertical line down from the top vertex to form a right angle.  This vertical line is .  Angle  for the supplementary (orange) triangle is .  Using the fact that , we can set up our formula to be the following:

 (either angle can be used and I will show this to be true)

  (Using from the original triangle)

This shows that when using this formula for an obtuse angle, you can use either the supplementary angle you made, or the original.  It is always helpful to draw this supplementary triangle in order to be able to visualize and understand logically how the formula is working for obtuse angles as well.

In order to use this formula you need to either the height (which can be used to find the angle) or the angle (which can be used to find the height).

The formula for the area of a triangle is (base)(height).  Since this is an obtuse triangle we need to break it into two right triangles by drawing the line down from the vertex perpendicular to the opposite side.

Now that we have two right triangles we can solve for the area of this triangle.  Notice our base is and our height is .  Plugging into the formula for the area of a triangle gives us .  Most of the time, we will not have an exact length for , but we may have, or will be able to solve for, the lengths of  and the angles .  Using our  relationship for right triangles, we know .  In this case we will use  as our angle.  So

We can plug  in for in our formula for area and we are left with .