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? Based on its angle measures, what kind of triangle is triangle ABC?

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Identify the known values: side \(a = \sqrt{5}\), side \(c = 2\sqrt{5}\), and angle \(A = 30^\circ\). We want to find \(\sin C\) and the measure of angle \(C\).

Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{c}{\sin C}\). This relates the sides and their opposite angles in any triangle.

Substitute the known values into the Law of Sines: \(\frac{\sqrt{5}}{\sin 30^\circ} = \frac{2\sqrt{5}}{\sin C}\).

Solve for \(\sin C\) by cross-multiplying and isolating \(\sin C\): \(\sin C = \frac{2\sqrt{5} \times \sin 30^\circ}{\sqrt{5}}\).

Once you find \(\sin C\), use the inverse sine function to determine angle \(C\): \(C = \sin^{-1}(\sin C)\). Then, analyze the angle measures to classify triangle \(ABC\) as acute, right, or obtuse based on the values of \(A\) and \(C\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Law of Sines

The Law of Sines relates the sides and angles of a triangle by stating that the ratio of a side length to the sine of its opposite angle is constant for all three sides. It is expressed as (a/sin A) = (b/sin B) = (c/sin C). This law is useful for finding unknown sides or angles in non-right triangles.

Sine Function and Angle Calculation

The sine function relates an angle in a triangle to the ratio of the length of the opposite side over the hypotenuse in right triangles, and is used in the Law of Sines for any triangle. Calculating sin C involves using known sides and angles, and finding angle C requires taking the inverse sine (arcsin) of the ratio obtained.

Classification of Triangles by Angles

Triangles are classified based on their angle measures: acute (all angles less than 90°), right (one angle exactly 90°), or obtuse (one angle greater than 90°). After finding angle C, the sum of angles A, B, and C helps determine the triangle type by comparing the measures to these categories.

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