Math / Trigonometry

QuestionInis question: 1 point(S) possible
Apply the law of sines to the following: $a=\sqrt{5}, \mathrm{c}=2 \sqrt{5}, \mathrm{~A}=30^{\circ}$ . What is the value of $\sin \mathrm{C}$ ? What is the measure of C ? Based on its angle measures, what kind of triangle is triangle ABC ?
What is the value of $\sin C$ ? $\square$ (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 $A B C$ ? Choose the correct answer below. Right Triangle Obtuse Triangle Acute Triangle

Studdy Solution

STEP 1

1. We are given a triangle $ABC$ with sides $a = \sqrt{5}$ , $c = 2\sqrt{5}$ , and angle $A = 30^\circ$ .
2. We will use the Law of Sines to find $\sin C$ and the measure of angle $C$ .
3. We will determine the type of triangle based on its angle measures.

STEP 2

1. Apply the Law of Sines to find $\sin C$ .
2. Calculate the measure of angle $C$ .
3. Determine the type of triangle $ABC$ .

STEP 3

Apply the Law of Sines, which states:
$\frac{a}{\sin A} = \frac{c}{\sin C}$
Substitute the known values:
$\frac{\sqrt{5}}{\sin 30^\circ} = \frac{2\sqrt{5}}{\sin C}$
Since $\sin 30^\circ = \frac{1}{2}$ , substitute this value:
$\frac{\sqrt{5}}{\frac{1}{2}} = \frac{2\sqrt{5}}{\sin C}$
Simplify the left side:
$2\sqrt{5} = \frac{2\sqrt{5}}{\sin C}$

STEP 4

Cross-multiply to solve for $\sin C$ :
$2\sqrt{5} \cdot \sin C = 2\sqrt{5}$
Divide both sides by $2\sqrt{5}$ :
$\sin C = 1$

STEP 5

Since $\sin C = 1$ , angle $C$ must be:
$C = 90^\circ$

STEP 6

Determine the type of triangle based on its angles. We have:
- $A = 30^\circ$ - $C = 90^\circ$
Since one angle is $90^\circ$ , triangle $ABC$ is a Right Triangle.
The value of $\sin C$ is:
$\boxed{1}$
The measure of $C$ is:
$\boxed{90^\circ}$
Based on its angle measures, triangle $ABC$ is a:
$\boxed{\text{Right Triangle}}$

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