When students work with right triangles they make use of trigonometric ratios. However, not all triangles can readily use the SOH CAH TOA trig ratios. This means that when attempting to solve the missing portion of non-right triangles we need to make use of different tools.
Law of Sines
The first tool used by students of geometry is the Law of Sines. This theorem reflects the relationship found in a triangle between the measure of the angles and the length of each opposing side.
If a student knows the measure of two angles and the length of one side they can use the Law Of Sines to solve the missing parts of the triangle. These scenarios are referred to as Angle-Angle-Side (AAS) and Angle-Side-Angle (ASA) respectively.
The Law of Sines also comes in handy if you know the length of two sides of the triangle, and the measure of one of the angles opposite to one of these sides. This scenario is described as the Side-Side-Angle (SSA) rule.
The Law of Sines is utilized by writing a proportion; students then perform a cross multiplication and solve the problem to determine the value of the missing piece of the ratio.
Law of Cosines
For times when the Law of Sines does not suffice, the Law of Cosines can frequently be used to solve a trinagle. The two major application for this theorem are SAS and SSS.
If the length of two sides is known - as well as the measure of the included angle – then you have a Side-Angle-Side (SAS) situation. This type of problem lends itself well to being solved by use of the Law of Cosines formula.
The other scenario where the Law of Cosines is useful is where the student knows the length of all three sides of the triangle. This situation is the Side-Side-Side (SSS) scenario; where the Law Of Cosines can be used to establish the measure of any or all of the three angles in the triangle.
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