Solving Triangles

Types of Triangle Information

In triangle problems, the type of information given (angles and sides) determines the method used to solve for unknowns. Recognizing the configuration is essential for selecting the correct approach.

• SAS (Side-Angle-Side): Two sides and the included angle are known. Use the Law of Cosines to solve for the unknown side or angle.

• AAA (Angle-Angle-Angle): All angles are known. No unique solution for side lengths; only the shape is determined.

• SSS (Side-Side-Side): All three sides are known. Use the Law of Cosines to find the angles.

• AAS (Angle-Angle-Side): Two angles and a non-included side are known. Use the Law of Sines to solve for unknowns.

• SSA (Side-Side-Angle): Two sides and a non-included angle are known. This is the ambiguous case; there may be zero, one, or two solutions.

Key Formulas

• Law of Sines:

• Law of Cosines:

Area of a Triangle

• SAS Formula:

• Heron's Formula (SSS):

Example Table: Triangle Information Types

Example: Solving a Triangle (SAS)

• Given: , ,

• Find using Law of Cosines:

• Calculate , then use Law of Sines to find other angles.

Solving Trigonometric Equations

General Approach

Trigonometric equations often require algebraic manipulation and the use of identities to find solutions within a specified interval.

• Interval: is commonly used for solutions in radians.

• Common Equations:

Key Trigonometric Identities

• Double Angle:

• Product-to-Sum:

• Sum-to-Product:

Example: Solving in

• Find all such that , etc.

• Divide by 2 to get , etc.

Example: Simplifying and Verifying Identities

• Simplify: Use the tangent addition formula:

• Verify: Express all terms in sine and cosine, then simplify.

Product-to-Sum and Sum-to-Product Formulas

Product-to-Sum

Sum-to-Product

Example: Write as a sum

• Use product-to-sum:

Summary Table: Triangle Solution Methods

Additional info:

• When solving triangles, always check if the sum of angles is .

• For SSA, check for the ambiguous case by comparing the given side to the height calculated from the known angle.

• Trigonometric equations may have multiple solutions within the interval .