Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 17-24: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), express the third part in terms of the first two. $$ \alpha, b ; \quad a $$

Step-by-Step Solution

Answer

\( a = \frac{b}{\cos(\alpha)} \)

Step 1: Understand the Given Parts

We are given a right-angled triangle \( \triangle ABC \) where \( \gamma = 90^{\circ} \). The parts provided are \( \alpha \) (angle \( A \)), \( b \) (side \( BC \)), and we need to express \( a \) (side \( AC \)) in terms of \( \alpha \) and \( b \).

Step 2: Use Trigonometric Ratios

In the right triangle, we use the cosine trigonometric ratio for angle \( \alpha \). The cosine of angle \( \alpha \) in \( \triangle ABC \) is defined as \( \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{a} \).

Step 3: Express \( a \) in Terms of \( \alpha \) and \( b \)

Rearrange the cosine equation to solve for \( a \). This gives us \( a = \frac{b}{\cos(\alpha)} \). Thus, we have expressed \( a \) in terms of \( \alpha \) and \( b \).

Key Concepts

Understanding a Right Triangle

A right triangle is a triangle where one angle is exactly 90 degrees. This specific angle, often represented by \( \gamma \), is known as the right angle. The triangle's sides are typically labeled: the side opposite the right angle is the hypotenuse, and the other two sides are known as the adjacent and opposite sides depending on the angle of interest. Understanding right triangles is crucial because they allow us to use special mathematical relationships called trigonometric ratios. These relationships are useful for solving various geometry and trigonometry problems, making them a fundamental part of mathematics.

• One side is always the hypotenuse, which is opposite the right angle and the longest side.
• The other two sides are called the legs of the triangle.
• All angles collectively sum to 180 degrees, so the other two angles must sum to 90 degrees.

Exploring Trigonometric Ratios

Trigonometric ratios are specific relationships of the angles and sides of a triangle. These ratios include sine, cosine, and tangent. For any right triangle and a particular angle \( \alpha \), these ratios define the relationship between two sides of the triangle:

• Sine (sin): the ratio of the length of the opposite side over the hypotenuse.
• Cosine (cos): the ratio of the length of the adjacent side over the hypotenuse.
• Tangent (tan): the ratio of the length of the opposite side over the adjacent side.

These relationships are invaluable tools in geometry. They allow us to find unknown side lengths or angles when two elements of the triangle are provided.

Understanding these ratios in-depth can greatly enhance problem-solving skills and make dealing with triangles much easier. Always remember that these ratios depend on the angle chosen, and knowing one angle besides the right angle can unlock the entire triangle's properties.

Using the Cosine Function

The cosine function focuses on the relationship between the adjacent side and the hypotenuse in a right triangle. In the case of triangle \( \triangle ABC \), where \( \gamma = 90^{\circ} \) and \( \alpha \) is not a right angle, cosine is expressed mathematically as \( \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} \). For this particular exercise, you are given the side \( b \) and angle \( \alpha \), and asked to find side \( a \) using these known quantities.

To solve for the hypotenuse \( a \), which in this scenario is the hypotenuse since angle \( \alpha \) is between side \( b \) and \( a \), rearrange the cosine equation:

• Start with \( \cos(\alpha) = \frac{b}{a} \).
• Rearrange to find the hypotenuse: \( a = \frac{b}{\cos(\alpha)} \).

This rearranged equation allows us to determine the length of the hypotenuse when the adjacent side and the angle are known. The simplicity and utility of the cosine ratio in expressing one side in terms of the other two aspects are why this function is widely used in solving problems related to right triangles.