Answer
• ∠B = 50º
,
• AB = 7.832
,
• CB = 5.035
Explanation
The triangle ABC is shown below:
As it is a right triangle, we can use trigonometric functions to solve it. To know the hypotenuse (AB), we can use the cosine function:
[tex]\cos(x)=\frac{adjacent\text{ side}}{hypotenuse}[/tex]
In our case, the adjacent side is b and the hypotenuse is AB. Then, by replacing our expressions we get:
[tex]\cos(A)=\frac{b}{AB}[/tex]
Next, by replacing the values and solving for AB we get:
[tex]\cos(40\degree)=\frac{6}{AB}[/tex][tex]AB=\frac{6}{\cos(40\degree)}[/tex][tex]AB\approx7.832[/tex]
As we have two sides, we can use the Pythagorean Theorem to find side CB:
[tex]AB^2=b^2+CB^2[/tex][tex]AB^2=b^2+CB^2[/tex]
Next, we can solve for CB (the side that we are lacking) as follows:
[tex]CB=\sqrt{AB^2-b^2}[/tex][tex]CB=\sqrt{7.832^2-6^2}[/tex][tex]CB=\sqrt{7.832^2-6^2}\approx5.035[/tex]
Finally, as the addition of the interior angles of a triangle adds up to 180º, we can find ∠B as follows:
[tex]\angle A+\angle B+\angle C=180\degree[/tex][tex]40\degree+\angle B+90\degree=180\degree[/tex][tex]\angle B=180\degree-90\degree-40\degree[/tex][tex]\angle B=50\degree[/tex]
