Question

apply the law of sines to the following: a = √5, c = 2√5, a = 30°. what is the value of sin c? what is the measure of c? (type an exact answer, using radicals as needed.) what is the measure of c? ° (type an integer or a decimal.) based on its angle measures, what kind of triangle is triangle abc? choose the correct answer below. acute triangle right triangle obtuse triangle

Explanation:

Part 1: Find \(\sin C\)

Step 1: Recall the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\). We know \(a = \sqrt{5}\), \(c = 2\sqrt{5}\), and \(A = 30^\circ\).

Step 2: Solve for \(\sin C\)

From \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we can re - arrange to get \(\sin C=\frac{c\sin A}{a}\).
Substitute the given values: \(a = \sqrt{5}\), \(c = 2\sqrt{5}\), and \(\sin A=\sin30^{\circ}=\frac{1}{2}\) into the formula.
\(\sin C=\frac{2\sqrt{5}\times\frac{1}{2}}{\sqrt{5}}\)
Simplify the numerator: \(2\sqrt{5}\times\frac{1}{2}=\sqrt{5}\)
Then \(\sin C=\frac{\sqrt{5}}{\sqrt{5}} = 1\)

Part 2: Find the measure of \(C\)

Step 1: Recall the range of sine function

We know that the range of the sine function \(y = \sin x\) is \([- 1,1]\), and \(\sin\theta=1\) when \(\theta = 90^{\circ}+360^{\circ}n\), \(n\in\mathbb{Z}\).
Since \(C\) is an angle in a triangle, \(0^{\circ}

Part 3: Determine the type of triangle

Step 1: Recall the definitions of triangle types

• An acute triangle has all angles less than \(90^{\circ}\).
• A right triangle has one angle equal to \(90^{\circ}\).
• An obtuse triangle has one angle greater than \(90^{\circ}\).

Since \(C = 90^{\circ}\), triangle \(ABC\) is a right triangle.

Answer:

s:

• The value of \(\sin C\) is \(1\).
• The measure of \(C\) is \(90^{\circ}\).
• Triangle \(ABC\) is a Right Triangle.