Problem 20 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The side \( a \) is given by \( a = b \times \tan(\alpha) \).

Step 1: Understand the given right triangle

In a right triangle, one of the angles is specifically known to be 90 degrees, in this case, angle \( \gamma = 90^\circ \). The sides of the triangle are usually referred to as opposites or adjacent based on the angle given. Here, you are given side \( b \), the angle \( \alpha \), and you are asked to find side \( a \).

Step 2: Identify the trigonometric relationship

By focusing on angle \( \alpha \), we can use trigonometric identities related to angle involvement in right triangles. For this scenario, side \( a \) is the opposite side of \( \alpha \), and side \( b \) is the adjacent side. The trigonometric function involving the opposite and adjacent side is tangent: \( \tan(\alpha) = \frac{a}{b} \).

Step 3: Solve for the unknown side \( a \)

Use the tangent ratio to express \( a \) in terms of known quantities. From the equation \( \tan(\alpha) = \frac{a}{b} \), solve for \( a \): \[ a = b \times \tan(\alpha) \] Now, \( a \) is expressed in terms of \( b \) and \( \alpha \).

Key Concepts

Tangent Ratio

The tangent ratio is a vital concept in trigonometry, especially when working with right triangles. It's defined through a specific angle in the triangle, often denoted as \( \alpha \) in the context of right triangle problems. The tangent of angle \( \alpha \) can be expressed as the ratio of the length of the opposite side to the adjacent side.

• Opposite Side: This is the side directly opposite the angle in question, which is angle \( \alpha \) in our case.
• Adjacent Side: This is the side next to the angle, but not the hypotenuse, which is directly opposite the right angle \( \gamma \).

Thus, the tangent function is defined by the equation:\[\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\]In practical terms, using the tangent ratio is incredibly useful for calculating unknown side lengths when you know one angle and one side of a right triangle. For example, when you know side \( b \) and angle \( \alpha \), you can conveniently find the opposite side \( a \) with \( a = b \times \tan(\alpha) \). This functionality makes the tangent ratio an indispensable tool for solving right triangles.

Right Triangles

Right triangles are a foundational topic in trigonometry and geometry. These triangles have one angle that is exactly \( 90^{\circ} \), which is known as the right angle. This feature leads to several unique properties and relationships worth understanding.

• Hypotenuse: The side opposite the right angle, and it is always the longest side in a right triangle.
• Legs: The two sides forming the right angle. These are often referred to as the 'opposite' and 'adjacent' sides concerning specific angles other than the \( 90^{\circ} \) angle.

The designation of opposite and adjacent depends on the angle within the triangle. For instance, if you focus on angle \( \alpha \), the side opposite \( \alpha \) becomes the opposite side, and the other leg becomes the adjacent side. These relationships are necessary when applying trigonometric properties, such as the tangent ratio, to find unknown side lengths or angles of a right triangle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any angle. They are critical tools for solving triangles and relating different trigonometric functions. One of the simplest yet powerful identities used in this context is the tangent identity.The basic trigonometric identities revolve around the sine, cosine, and tangent functions. Specifically, for any angle \( \alpha \) in a right triangle:\[\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}\]This identity connects tangent to the more familiar sine and cosine functions, making it easier to derive or cross-verify results. Knowing these relationships can help solve various problems, like finding unknown side lengths or verifying the correctness of an angle's tangent value.Understanding and applying these identities allow students to approach trigonometric problems with confidence, knowing they have robust equations to verify and solve the various relationships between angles and sides in a right triangle.