Answer:
The remaining side and angles are [tex]a = 20\sqrt{3}[/tex], [tex]\angle A = 60^{\circ}[/tex] and [tex]\angle B = 30^{\circ}[/tex].
Step-by-step explanation:
According to the information given on statement, we are in front of a right triangle because [tex]\angle C[/tex], opossite to side [tex]c[/tex], is a right angle. Hence, [tex]c[/tex] is the hypotenuse of the right triangle and [tex]b[/tex], a leg. The missing length can be calculated by the Pythagorean Theorem:
[tex]a = \sqrt{c^{2}-b^{2}}[/tex] (1)
If we know that [tex]c = 40[/tex] and [tex]b = 20[/tex], then the length of the missing leg is:
[tex]a = \sqrt{c^{2}-b^{2}}[/tex]
[tex]a = 20\sqrt{3}[/tex]
Lastly, we find the value of the missing angles by means of direct and inverse trigonometric relations:
Angle A
[tex]\angle A = \tan^{-1}\left(\frac{a}{b} \right)[/tex] (2)
Angle B
[tex]\angle B = \tan^{-1} \left(\frac{b}{a} \right)[/tex] (3)
If we know that [tex]a = 20\sqrt{3}[/tex] and [tex]b = 20[/tex], then the values of the missing angles are, respectively:
Angle A
[tex]\angle A = \tan^{-1}\left(\frac{a}{b} \right)[/tex]
[tex]\angle A = 60^{\circ}[/tex]
Angle B
[tex]\angle B = \tan^{-1} \left(\frac{b}{a} \right)[/tex]
[tex]\angle B = 30^{\circ}[/tex]
The remaining side and angles are [tex]a = 20\sqrt{3}[/tex], [tex]\angle A = 60^{\circ}[/tex] and [tex]\angle B = 30^{\circ}[/tex].
