Answer:
28.31
Step-by-step explanation:
So in this case you're going to need to use the law of sines which essentially states that: [tex]\frac{a}{sinA}=\frac{b}{sinB}[/tex] which should apply to any of the sides. the lowercase a and b are the opposite sides of the angles A and B. In this example the angle A is given and C is given, but not in the text. Since you have the right angle thing in the diagram, angle C is a right angle (90 degrees).
Given information:
[tex]\angle A = 32\\a=15\\\angle C = 90 \text{(the right angle symbol in the diagram of the triangle)}[/tex]
Law of sines equation:
[tex]\frac{a}{sinA}=\frac{b}{sinB} \text{ can be applied to any two sides }[/tex]
Plug in known information:
[tex]\frac{15}{sin(32)}=\frac{c}{sin (90)}[/tex]
since sin(90) = 1, simplify the fraction
[tex]\frac{15}{sin(32)} = c[/tex]
Calculate sin(32)
[tex]\frac{15}{0.530}\approx c[/tex]
Divide
[tex]28.306\approx c[/tex]
If you haven't learned law of sines yet and don't quite understand why it works you can also use the definition of tan to find what b equals and then use the Pythagorean Theorem to solve for c
tan is defined as: [tex]\frac{opposite}{adjacent}[/tex]
[tex]tan(32)=\frac{15}{b}[/tex]
now multiply both sides by b
[tex]b * tan(32)=15[/tex]
Divide both sides by tan 32
[tex]b = \frac{15}{tan(32)}[/tex]
[tex]b\approx 24.005[/tex]
Now use the Pythagorean Theorem: [tex]a^2+b^2=c^2[/tex]
[tex](24.005)^2+15^2=c^2\\[/tex]
Square known values
[tex]576.241+225=c^2[/tex]
Add the values
[tex]801.241 = c^2[/tex]
take the square root of both sides
[tex]28.306\approx c[/tex]
Rounding to the nearest hundredth gives you 28.31
