In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. a = 5, b = 7, c = 10

Verified step by step guidance

1. Identify the type of triangle using the given side lengths: a = 5, b = 7, c = 10. Since the sum of the squares of the two shorter sides (a^2 + b^2) is less than the square of the longest side (c^2), this is an obtuse triangle.
2. Use the Law of Cosines to find one of the angles. For example, to find angle C, use the formula: $c^2=a^2+b^2-2ab\cdot \cos (C)$. Substitute the given values into the formula and solve for $\cos (C)$.
3. Calculate angle C using the inverse cosine function: $C={\cos }^{-1}(\text{calculated value})$. Round the result to the nearest degree.
4. Use the Law of Sines to find another angle. For example, to find angle A, use the formula: $\frac{a}{\sin (A)}=\frac{c}{\sin (C)}$. Substitute the known values and solve for $\sin (A)$.
5. Calculate angle A using the inverse sine function: $A={\sin }^{-1}(\text{calculated value})$. Round the result to the nearest degree. Finally, find angle B by using the fact that the sum of angles in a triangle is 180 degrees: $B={180}^{\circ }-A-C$.

Key Concepts

Law of Cosines

The Law of Cosines is a fundamental formula used in trigonometry to relate the lengths of the sides of a triangle to the cosine of one of its angles. It states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively, the formula is c² = a² + b² - 2ab * cos(C). This law is particularly useful for solving triangles when two sides and the included angle are known or when all three sides are known.

Law of Sines

The Law of Sines is another essential theorem in trigonometry that relates the ratios of the lengths of the sides of a triangle to the sines of its angles. It states that (a/sin(A)) = (b/sin(B)) = (c/sin(C)). This law is particularly useful for solving triangles when two angles and one side are known or when two sides and a non-included angle are known, allowing for the determination of unknown angles and sides.

Triangle Properties

Understanding the properties of triangles, including the sum of angles and the relationship between sides and angles, is crucial for solving triangle problems. The sum of the interior angles in any triangle is always 180 degrees. Additionally, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. These properties help in determining unknown angles and sides when applying the Law of Cosines or the Law of Sines.