Problem 18 - 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. $$ \beta, c ; \quad b $$

Step-by-Step Solution

Answer

\( b = c \cdot \sin(\beta) \)

Step 1: Understand Given Parts of Triangle

We are given a right triangle \( \triangle ABC \) with \( \gamma = 90^\circ \), and we need to express side \( b \) in terms of angle \( \beta \) and hypotenuse \( c \).

Step 2: Use Trigonometric Ratios

In a right triangle, the side opposite to \( \beta \) (angle \( B \)) is \( b \), and the hypotenuse is \( c \). So, use the sine function: \[\sin(\beta) = \frac{b}{c}\].

Step 3: Solve for Side b

Rearrange the equation \( \sin(\beta) = \frac{b}{c} \) to solve for \( b \): \[b = c \cdot \sin(\beta)\].

Key Concepts

Understanding Right Triangles

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees, also known as a right angle. This angle is typically denoted by the Greek letter \( \gamma \) in mathematical problems and diagrams. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides are known as the 'legs' of the triangle.

• The side opposite an angle in a right triangle is crucial in defining trigonometric functions.
• Usually, problems involving right triangles will give you certain values, such as the length of a side or the measure of an angle, and from there you will solve for the unknown values.
• It's important to remember that the right angle always makes an angle of 90 degrees.

Knowing the properties of right triangles can help simplify and solve many math problems related to angles and side lengths.

Delving into Trigonometric Ratios

Trigonometric ratios are mathematical relationships used in right triangles to relate angles to side lengths. There are three primary trigonometric ratios: sine, cosine, and tangent. Each of these ratios is defined based on a given angle of a right triangle.

• The sine ratio (sin) is the ratio of the length of the side opposite an angle to the length of the hypotenuse.
• The cosine ratio (cos) is the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
• The tangent ratio (tan) is the ratio of the length of the side opposite the angle to the side adjacent to it.

These ratios are fundamental in solving for unknown sides or angles in right triangles. By knowing an angle and a side length, you can use these ratios to calculate other unknown parts of the triangle. This process is central to understanding and applying trigonometry.

Exploring the Sine Function

The sine function is a key concept in trigonometry. It is particularly useful when working with right triangles, as it helps in connecting angles to side lengths. The sine of an angle \( \beta \) in a right triangle is given by the formula:
\[ \sin(\beta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]
In the context of our problem, the sine function helps express the side \( b \), which is opposite to angle \( \beta \), in terms of the hypotenuse \( c \). By rearranging the equation \( \sin(\beta) = \frac{b}{c} \), we find:
\[ b = c \cdot \sin(\beta) \]

• This formula is very useful as it highlights how you can find one side of a right triangle if you know the angle and the hypotenuse.
• The sine function is directly related to the unit circle and also finds applications beyond right triangles, such as in waves and oscillations in physics.

By mastering the sine function, you can solve a wide variety of practical and theoretical problems in mathematics and science.