Solution of Triangles: Projection Formula

The solution of triangles is an important aspect of trigonometry that deals with the determination of the remaining elements of a triangle when some of its elements are known. The projection formula is a set of equations that relate the sides and angles of a triangle. It is particularly useful in solving oblique triangles, which are non-right triangles.

The Projection Formula

The projection formula is derived from the law of cosines and relates the lengths of the sides of a triangle to the projection of one side onto another. For a triangle with sides a, b, and c, and angles A, B, and C opposite to these sides respectively, the projection formula is given by:

[ b = c \cos(A) + a \cos(B) ] [ c = a \cos(B) + b \cos(C) ] [ a = b \cos(C) + c \cos(A) ]

These formulas can be understood as the projection of one side onto another when the triangle is split into two right-angled triangles.

Important Points and Differences

Here is a table summarizing the important points and differences related to the projection formula:

Aspect	Description
Applicability	Used for non-right triangles (oblique triangles).
Relation to Other Formulas	Derived from the law of cosines.
Use	Helps in finding unknown sides or angles when certain elements of a triangle are known.
Components	Involves both the sides and angles of a triangle.

Examples

Let's look at some examples to understand how the projection formula is applied.

Example 1: Finding the Length of a Side

Given a triangle with sides a = 7, b = 5, and angle C = 60°, find the length of side c.

Using the projection formula:

[ c = a \cos(B) + b \cos(C) ]

Since we don't have angle B, we can use the fact that the sum of angles in a triangle is 180° to find angle B.

[ B = 180° - C - A ]

Assuming angle A is also known or can be calculated, we can find B and then use the cosine of B and C to find c.

Example 2: Finding the Length of a Side with Given Angles

Given a triangle with sides a = 8, b = 6, and angles A = 45° and B = 60°, find the length of side c.

First, we find angle C:

[ C = 180° - A - B = 180° - 45° - 60° = 75° ]

Now, using the projection formula:

[ c = a \cos(B) + b \cos(C) ] [ c = 8 \cos(60°) + 6 \cos(75°) ] [ c = 8 \times \frac{1}{2} + 6 \times (\text{cosine of } 75°) ] [ c = 4 + 6 \times (\text{cosine of } 75°) ]

After calculating the cosine of 75°, we can find the exact length of side c.

Conclusion

The projection formula is a powerful tool in the solution of triangles, allowing us to find unknown sides or angles given certain known elements of a triangle. It is particularly useful when dealing with oblique triangles, where the standard right triangle trigonometric ratios do not apply directly. Understanding and applying the projection formula can greatly simplify the process of solving complex trigonometric problems.