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Chapter 1: Problem 1

Define the six trigonometric functions in terms of the sides of a right triangle.

Short Answer

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Question: Define the six trigonometric functions in terms of the sides of a right triangle and their relationships with each other. Answer: The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). They are defined in terms of the sides of a right triangle as follows: 1. Sin(胃) = opposite/hypotenuse 2. Cos(胃) = adjacent/hypotenuse 3. Tan(胃) = opposite/adjacent 4. Csc(胃) = hypotenuse/opposite 5. Sec(胃) = hypotenuse/adjacent 6. Cot(胃) = adjacent/opposite The relationships between the primary and reciprocal trigonometric functions are: 1. Csc(胃) = 1/sin(胃) 2. Sec(胃) = 1/cos(胃) 3. Cot(胃) = 1/tan(胃)

Step by step solution

01

Understand the basic terms used in right triangles

In a right triangle, we have three sides: 1. Hypotenuse: The longest side, which is always opposite to the right angle. 2. Opposite: The side that is opposite to the angle of interest. 3. Adjacent: The side that forms the angle of interest with the hypotenuse. We will use these three sides to define the six trigonometric functions.
02

Define the primary trigonometric functions

The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). They can be defined as follows for an angle 胃 in the right triangle: 1. Sine (sin): \(\sin{\theta} = \frac{opposite}{hypotenuse}\) 2. Cosine (cos): \(\cos{\theta} = \frac{adjacent}{hypotenuse}\) 3. Tangent (tan): \(\tan{\theta} = \frac{opposite}{adjacent}\)
03

Define the reciprocal trigonometric functions

The reciprocal trigonometric functions are related to the primary trigonometric functions, and they are cosecant (csc), secant (sec), and cotangent (cot). They can be defined as follows for an angle 胃 in the right triangle: 1. Cosecant (csc): \(\csc{\theta} = \frac{1}{\sin{\theta}} = \frac{hypotenuse}{opposite}\) 2. Secant (sec): \(\sec{\theta} = \frac{1}{\cos{\theta}} = \frac{hypotenuse}{adjacent}\) 3. Cotangent (cot): \(\cot{\theta} = \frac{1}{\tan{\theta}} = \frac{adjacent}{opposite}\) Now, we have defined all six trigonometric functions in terms of the sides of a right triangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right Triangle
In the world of geometry, the right triangle stands out due to its remarkable simplicity and utility. A right triangle is a triangle in which one of the angles is exactly 90 degrees. This special feature allows us to explore various geometric concepts and trigonometric functions.

Every right triangle consists of three sides:
  • Hypotenuse: This is the longest side of the triangle, opposite the right angle. It's the most significant side when defining trigonometric functions.
  • Opposite: This is the side opposite the angle of interest or the angle we are examining, except the right angle.
  • Adjacent: This is the side that lies next to the angle of interest. Along with the hypotenuse, it forms the right angle.
Understanding the layout of these three sides is crucial when working with trigonometric functions. It's the relationship between these sides that forms the essence of trigonometry, allowing us to solve problems involving angles and lengths in a right triangle.
Sine Cosine Tangent
The primary trigonometric functions are the backbone of trigonometry. They are Sine (sin), Cosine (cos), and Tangent (tan). These functions help in understanding and solving problems related to angles and side lengths in right triangles.

Here is how each function is defined:
  • Sine (sin θ): This function is the ratio of the length of the opposite side to the hypotenuse. It is expressed as \[\sin{ heta} = \frac{\text{opposite}}{\text{hypotenuse}}.\]
  • Cosine (cos θ): This is the ratio of the adjacent side to the hypotenuse, given by \[\cos{ heta} = \frac{\text{adjacent}}{\text{hypotenuse}}.\]
  • Tangent (tan θ): This function compares the opposite side to the adjacent side, shown by \[\tan{ heta} = \frac{\text{opposite}}{\text{adjacent}}.\]
These functions are foundational in trigonometry and find applications in various real-world scenarios, such as calculating heights, distances, and more.
Reciprocal Trigonometric Functions
The reciprocal trigonometric functions are integral to advanced trigonometric calculations and complement the primary functions. They include Cosecant (csc), Secant (sec), and Cotangent (cot). These functions are essentially the reciprocal of sin, cos, and tan, respectively.

The reciprocal relationships are as follows:
  • Cosecant (csc θ): Inverse of sine, defined as \[\csc{ heta} = \frac{1}{\sin{\theta}} = \frac{\text{hypotenuse}}{\text{opposite}}.\]
  • Secant (sec θ): Inverse of cosine, given by \[\sec{ heta} = \frac{1}{\cos{\theta}} = \frac{\text{hypotenuse}}{\text{adjacent}}.\]
  • Cotangent (cot θ): Inverse of tangent, expressed as \[\cot{ heta} = \frac{1}{\tan{\theta}} = \frac{\text{adjacent}}{\text{opposite}}.\]
These reciprocal functions extend the power of trigonometry by providing alternate ways to approach and solve problems in right triangles. They are especially helpful when working with angles and calculations involving the hypotenuse.

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