Question

apply the law of sines to the following: ( a = sqrt{5} ), ( c = 2sqrt{5} ), ( a = 30^circ ). what is the value of ( sin c )? what is the measure of ( c )? based on its angle measures, what kind of triangle is triangle ( abc )?
1 (type an exact answer, using radicals as needed.)
what is the measure of ( c )?
( square^circ ) (type an integer or a decimal.)

Explanation:

Step1: State the Law of Sines

$\frac{a}{\sin A} = \frac{c}{\sin C}$

Step2: Substitute given values

$\frac{\sqrt{5}}{\sin 30^\circ} = \frac{2\sqrt{5}}{\sin C}$

Step3: Simplify $\sin 30^\circ$

$\sin 30^\circ = \frac{1}{2}$, so left side is $\frac{\sqrt{5}}{1/2} = 2\sqrt{5}$

Step4: Solve for $\sin C$

$2\sqrt{5} = \frac{2\sqrt{5}}{\sin C} \implies \sin C = \frac{2\sqrt{5}}{2\sqrt{5}} = 1$

Step5: Find measure of $C$

$\sin C = 1 \implies C = 90^\circ$

Answer:

1
90°