Problem 18 - Algebra 2 and Trigonometry

Question

Use the Law of Sines to show that if \(\angle C\) of \(\triangle A B C\) is a right angle, \(\sin A=\frac{a}{c}\) .

Step-by-Step Solution

Answer

Using the Law of Sines for a right triangle, \( \sin A = \frac{a}{c} \).

Step 1: Given Information

We have a triangle \( \triangle ABC \) with \( \angle C \) as a right angle. We need to use the Law of Sines to prove that \( \sin A = \frac{a}{c} \) where \( a = BC \), \( b = AC \), and \( c = AB \).

Step 2: State the Law of Sines

The Law of Sines states that for any triangle, \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). In this problem, \( \angle C \) is a right angle, therefore \( \sin C = 1 \) since \( \sin 90^\circ = 1 \). This simplifies one part of the Law of Sines for right triangle ABC.

Step 3: Apply the Law of Sines to \( \triangle ABC \)

For \( \angle C \) as a right angle, the Law of Sines becomes: \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Simplifying further gives \( \frac{a}{\sin A} = \frac{c}{1} \).

Step 4: Rearrange to Find \( \sin A \)

From \( \frac{a}{\sin A} = c \), we can rearrange to isolate \( \sin A \). Multiply both sides by \( \sin A \) and divide both sides by \( c \) to get \( \sin A = \frac{a}{c} \). Thus, we have shown the required identity.

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles measures exactly 90 degrees. This is why we call it a "right" angle. In a right triangle, this particular angle is often denoted as \( \angle C \). As a result, the side opposite this angle, known as the hypotenuse, is the longest side of the triangle.

• Every right triangle consists of one right angle and two other angles that are complementary, meaning they add up to 90 degrees.
• The sides of a right triangle are usually labeled as \( a \) and \( b \) for the two shorter sides, while \( c \) is used for the hypotenuse.

Understanding the properties of right triangles is essential in trigonometry, as they form the foundation for defining the sine, cosine, and tangent functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equality are defined. They are derived from the geometric relations inherent in right triangles.

• For right triangles, the Pythagorean identity \( \sin^2 A + \cos^2 A = 1 \) is fundamental.
• Another essential identity is \( \tan A = \frac{\sin A}{\cos A} \), which connects the sine, cosine, and tangent functions.

These identities are not just abstract rules; they express relationships that allow us to solve problems and find unknown angles or sides of triangles. In our exercise, the identity \( \sin C = 1 \) for a right triangle (where \( C \) is \( 90^\circ \)) plays a crucial role in simplifying the Law of Sines.

Sine Function

The sine function, usually denoted as \( \sin \), is a fundamental trigonometric function used extensively in geometry and trigonometry. It measures the ratio of the length of the opposite side to the hypotenuse in a right triangle.

• The sine function is periodic and oscillates between -1 and 1.
• In a right triangle, \( \sin A = \frac{\text{opposite side}}{\text{hypotenuse}} \).
• It reaches its maximum value at \( 90^\circ \), where \( \sin 90^\circ = 1 \).

For triangle \( \triangle ABC \) with \( \angle C \) as a right angle, \( \sin A = \frac{a}{c} \) can be easily derived using the Law of Sines. This relationship between sides and angles is pivotal in trigonometry, making the sine function indispensable for solving and understanding trigonometric problems.