Problem 17 - Algebra and Trigonometry

Question

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=5, b=7, c=10$$

Step-by-Step Solution

Answer

Using the Law of Cosines, we first find angle C, then use the Law of Sines to find another angle, either A or B. Finally, the remaining angle is calculated by subtracting the sum of the two known angles from 180. After calculations, make sure to round lengths to the nearest tenth and angle measures to the nearest degree as required.

Step 1: Use the law of cosines to find an angle

Given three sides of a triangle, the law of cosines can be used to find an angle. This below is the law of cosines: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] After substitute a, b, c into this formula, we can solve for angle C.

Step 2: Use the Law of Sines to find another angle

After finding angle C, the Law of Sines can be applied to find another angle. The formula is: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] Given angle C and sides a, b, c, we can solve for angle A or B using this formula.

Step 3: Find the last angle

We know that the sum of all angles in a triangle is 180 degrees. After the two angles have be found, the third angle can be calculated by subtract the sum of two angles from 180. \[ Degree\_missing = 180 - (Degree\_found\_1 + Degree\_found\_2) \]