Solve each triangle. See Examples 2 and 3.


A = 80° 40', b = 143 cm, c = 89.6 cm

Verified step by step guidance

1. Convert the given angle A from degrees and minutes to decimal degrees for easier calculation. Recall that 1 minute is \( \frac{1}{60} \) of a degree, so calculate \( A = 80 + \frac{40}{60} \) degrees.
2. Use the Law of Cosines to find side \( a \). The Law of Cosines formula is: \[ a^2 = b^2 + c^2 - 2bc \cos(A) \]. Substitute the known values of \( b \), \( c \), and \( A \) into this formula.
3. Calculate \( a^2 \) using the substituted values, then take the square root to find \( a \). This gives the length of side \( a \).
4. Next, use the Law of Sines to find another angle, for example angle \( B \). The Law of Sines states: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \]. Rearrange to solve for \( \sin(B) \): \[ \sin(B) = \frac{b \sin(A)}{a} \].
5. Calculate \( \sin(B) \) using the values of \( a \), \( b \), and \( A \), then find angle \( B \) by taking the inverse sine (arcsin). Finally, find angle \( C \) using the fact that the sum of angles in a triangle is 180°, so \( C = 180° - A - B \).

Key Concepts

Law of Cosines

The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is especially useful for solving triangles when two sides and the included angle are known, allowing calculation of the third side or other angles.

Angle Conversion and Notation

Angles given in degrees and minutes (e.g., 80° 40') must be converted to decimal degrees or radians for calculation. Understanding this notation ensures accurate use of trigonometric functions and prevents errors in solving the triangle.

Triangle Solution Strategy

Solving a triangle involves finding all unknown sides and angles using given data. Typically, start with known angles and sides, apply the Law of Cosines or Law of Sines as appropriate, and verify results to ensure the triangle's angle sum equals 180°.