Chapter 8.1, Problem 33AYU

In Problems 29-42, use the right triangle shown below. Then, using the given information, solve the triangle.

$b=4$, $A={10}^{\circ }$; find a, c, and B

To determine

To find: The values of $a$, $c$, and $\text{B}$.

Answer to Problem 33AYU

$\text{A }={80}^{\circ },c\approx 4.06,a\approx 0.71$

Explanation of Solution

Given:

$b=4,\text{ A }=\text{ 1}{0}^{\circ }$;

Formula used:

$\cos \theta \text{ = }\frac{\text{adjacent side}}{\text{hypotenuse}}$

$\tan \theta \text{ = }\frac{\text{opposite side}}{\text{adjacent side}}$ and Sum of the interior angles in a triangle is ${180}^{\circ }$.

Calculation:

We know that $\text{A }=\text{ 1}{0}^{\circ }$ and it’s a right angle triangle that means $\text{C }=\text{ 9}{0}^{\circ }$.

So $\text{A + B + C }={180}^{\circ }$

${10}^{\circ }+\text{ B }+\text{ 9}{0}^{\circ }={180}^{\circ }$

$\text{B }+{\text{ 100}}^{\circ }={180}^{\circ }$

$\text{B }={180}^{\circ }-{\text{ 100}}^{\circ }$

$\text{B }={\text{ 80}}^{\circ }$

From the figure,

$\cos 1{\text{0}}^{\circ }\text{ = }\frac{\text{adjacent side}}{\text{hypotenuse}}$

$\cos 1{\text{0}}^{\circ }\text{ = }\frac{b}{c}$

$\cos 1{\text{0}}^{\circ }\approx 0.9848$

$0.9848=\frac{4}{c}$

$c\text{ = }\frac{4}{0.9848}$

$c\approx 4.06$

$\tan {10}^{\circ }\text{ = }\frac{\text{opposite side}}{\text{adjacent side}}$

$\tan {10}^{\circ }=\frac{a}{b}$

$\tan {10}^{\circ }\approx 0.1763$

$0.1763=\frac{a}{4}$

$a=0.1763\times 4$

$a\approx 0.71$