Chapter 8.3, Problem 26AYU

In Problems 17-32, solve each triangle.

$a\text{ = 4}$, $b\text{ = 5}$, $c\text{ = 3}$

To determine

To find: The remaining angle(s) and side(s) of the given triangle.

Answer to Problem 26AYU

Solution:

$A \approx 53.1°, B \approx 90° and C \approx 36.9°$

Explanation of Solution

Given:

$a = 4, b = 5, c = 3$

Formula used:

$b² = a² + c² - 2ac\text{cos}B$

$c² = a² + b² - 2ab\text{cos}C$

Sum of the interior angles in a triangle is $180°$

Calculation:

$a = 4, b = 5, c = 3$

$b² = a² + c² - 2ac\text{cos}B$

$5^2 = 4² + 3² - 2 * \left(4\right)\left(3\right)\text{cos}B$

$\text{cos}B = \frac{16 - 9 - 25}{24}$

$B = 90°$

$a² = b² + c² - 2bc\text{cos}A$

$\text{cos}A = \frac{b^2 + c² - a²}{2bc}$

$\text{cos}A = \frac{5^2 + 3^2 - 4^2}{2 * 3 * 5}$

$A = {\text{cos}}^{ - 1}\left(0.6\right) \approx 53.1°$

$A + B + C = 180°$

$C = 180° - 90° - 53.1°$

$C \approx 36.9°$