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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.