Non-right triangles

So far, we've only dealt with right triangles, but trigonometry can be easily applied to non-right triangles because any non-right triangle can be divided by an altitude* into two right triangles.

Roll over the triangle to see what that means →

 

* An altitude is a line segment that has one endpoint at a vertex of a triangle intersects the opposite side at a right angle. See triangles.

Labeling of non-right triangles

This labeling scheme is commonly used for non-right triangles. Capital letters are angles and the corresponding lower-case letters go with the side opposite the angle: side a (with length of a units) is across from angle A (with a measure of A degrees or radians), and so on.

labeling of NR triangles

law of sines derivation

 

The law of sines

Consider the triangle on the left. if we find the sines of angle A and angle C using their corresponding right triangles, we notice that they both contain the altitude, x.

Solving for the altitude lets us use the transitive property: If a·sin(A) = x and c·sin(C) = x, then a·sin(A) must be equal to c·sin(C).

That leads directly to the law of sines.

We could do the same derivation with the other two altitudes, drawn from angles A and C to come up with similar relations for the other angle pairs. We call these together the law of sines.

The law of sines can be used to find the measure of an angle or a side of a non-right triangle if we know:

  • two sides and an angle not between them or
  • two angles and a side not between them.

law of sines

 

Examples: Law of sines

Use the law of sines to find the missing measurements of the triangles in these examples. In the first, two angles and a side are known. In the second two sides and an angle. Notice that we need to know at least one angle-opposite side pair for the Law of Sines to work.

Example 1: Find all of the missing measurements of this triangle:

LOS example 1

 

Example 2: Find all of the missing measurements of this triangle:

LOS example 2

 

LOC derivation

The law of cosines

← Consider another non-right triangle, labeled as shown with variable lengths x and y. We can derive a similar law containing the cosine function.

First use the Pythagorean theorem to derive two equations for each right triangle. Notice that each contains and x2, so you can eliminate that between the two using the transitive property.

Then do a little algebra to get to

c2 = a2 - b2 + 2by,

which can be rearranged to:

a2 = b2 + c2 - 2by.

Then incorporate the cosine function for the y-variable as shown to get the Law of Cosines

The law of cosines can be used to find the measure of an angle or a side of a non-right triangle if we know:

  • two sides and the angle between them or
  • three sides and no angles.

We could again do the same derivation using the other two altitudes of our triangle, to yield three versions of the law of cosines for any triangle. They are listed below.

law of cosines

 

The Law of Cosines is just the Pythagorean relationship with a correction factor, -2bc·cos(A), to account for the fact that the triangle is not a right triangle.

 

Examples: Law of cosines

Use the law of cosines to find the missing measurements of the triangles in these two examples. In the first, the measures of two sides and the included angle (the angle between them) are known. In the second, three sides are known.

Example 1: Find all of the missing measurements of this triangle:

LOC example 1

 

Example 2: Find all of the missing measurements of this triangle:

LOC example 2

 

under construction 1

 

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