Problem 49 - Thinking Mathematically

Question

If you are given the lengths of the sides of a right triangle, describe how to find the sine of either acute angle.

Step-by-Step Solution

Answer

The sine of an acute angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse of the triangle.

Step 1: Understand the Triangle

A right triangle has three sides: the hypotenuse, which is the longest side, and the two other sides known as the adjacent and opposite sides. The hypotenuse is always opposite the right angle, and the adjacent and opposite sides are relative to the angle of interest.

Step 2: Identify the Acute Angle and Opposite Side

Choose one of the acute angles (not the 90-degree angle). The side that is directly across from this chosen angle is the opposite side for that angle.

Step 3: Identify the Hypotenuse

The hypotenuse is the longest side of a right triangle, and it is always opposite the right angle. It doesn't change even if we change the acute angle of interest.

Step 4: Calculate the Sine

Given the right triangle, the sine of an acute angle (\(theta\)) is calculated using the formula \( sin(theta) = \frac{Opposite}{Hypotenuse} \). Substitute the given lengths of the opposite side and the hypotenuse into the formula and perform the division to calculate the sine of a given angle.

Key Concepts

Right Triangle

A right triangle is a fundamental shape in trigonometry with unique properties that make it distinct. It consists of three sides: the hypotenuse, and two legs known as the adjacent and the opposite sides relative to an angle. The hypotenuse is the longest side and is always found opposite the right angle, which measures 90 degrees. In a right triangle, the right angle defines it, and the two remaining angles are acute, meaning they are less than 90 degrees. These acute angles are crucial for calculating trigonometric functions. When working with a right triangle, it's essential to understand which angle and side are being referenced, as this affects the calculation and understanding of trigonometric ratios.

Sine Formula

The sine formula is one of the basic trigonometric functions used to establish relationships between angles and sides in a right triangle. The sine of a given acute angle, typically represented as \( \sin(\theta) \), is calculated by taking the ratio of the length of the opposite side to the hypotenuse. Mathematically, this is expressed as:

• \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

This formula helps in determining unknown properties of a triangle when certain sides are known. It's an essential part of solving problems related to right triangles and is widely used in various fields such as physics, engineering, and architecture to calculate angles and distances.

Acute Angle Identification

In the context of a right triangle, identifying an acute angle is crucial for applying trigonometric ratios effectively. An acute angle is any angle that is less than 90 degrees. In a right triangle, besides the right angle, the other two angles are acute. These are the angles where trigonometric properties are typically computed. To identify an acute angle:

• Look for the angle which is not the right angle (90 degrees).
• Ensure that the sum of all angles in the triangle equals 180 degrees, confirming the presence of acute angles.

Once the acute angle is recognized, it can determine which side is opposite and which side is adjacent, crucial for calculating sine, cosine, or tangent ratios.

Hypotenuse

The hypotenuse of a right triangle is the side opposite the right angle, and it holds special significance due to its role in trigonometric calculations. Being the longest side, the hypotenuse serves as the denominator in the sine formula and other trigonometric ratios, linking the lengths of sides to the angles within the triangle. To determine which side is the hypotenuse:

• Identify the side that is opposite the 90-degree angle.
• Remember that the hypotenuse is always the triangle's longest side.

Understanding and correctly identifying the hypotenuse is vital for accurate application of the sine formula, as it directly affects the magnitude of sine, cosine, and tangent values used in various calculations.