Problem 14 - Algebra 2

Question

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((7,24)\)

Step-by-Step Solution

Answer

The six trigonometric functions are: \(\sin\theta = \frac{24}{25}\), \(\cos\theta = \frac{7}{25}\), \(\tan\theta = \frac{24}{7}\), \(\csc\theta = \frac{25}{24}\), \(\sec\theta = \frac{25}{7}\), \(\cot\theta = \frac{7}{24}\).

Step 1: Understand the Problem

We need to find the six trigonometric functions for an angle \(\theta\) in standard position where the terminal side of \(\theta\) passes through the point (7, 24). The angle \(\theta\) can be visualized in the coordinate plane, where (7, 24) is on this terminal side.

Step 2: Calculate the Radius

The radius \(r\) in the coordinate plane is calculated using the formula \(r = \sqrt{x^2 + y^2}\). For the point (7, 24), this calculates as \[r = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25\].

Step 3: Define Trigonometric Functions

Identify the basic trigonometric ratios using \(x = 7\), \(y = 24\), and \(r = 25\):- \(\sin\theta = \frac{y}{r} = \frac{24}{25}\)- \(\cos\theta = \frac{x}{r} = \frac{7}{25}\)- \(\tan\theta = \frac{y}{x} = \frac{24}{7}\).

Step 4: Calculate Reciprocal Functions

From the basic trigonometric functions, calculate their reciprocals:- \(\csc\theta = \frac{1}{\sin\theta} = \frac{25}{24}\)- \(\sec\theta = \frac{1}{\cos\theta} = \frac{25}{7}\)- \(\cot\theta = \frac{1}{\tan\theta} = \frac{7}{24}\).

Key Concepts

Standard Position

In trigonometry, understanding the concept of an angle in "standard position" is essential. An angle is said to be in standard position when its vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis. The "terminal side" of the angle is where it ends after rotation.
This concept is crucial because it allows the use of coordinates to determine trigonometric functions. When given a point like (7, 24), it lies on the terminal side of the angle. The relationship between the coordinates of this point and the trigonometric functions can be leveraged to find exact values.
It is important to visualize standard position when solving problems like these, as it sets a uniform way to reference any angle by its intersection with the coordinate plane.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is used to graphically represent points, lines, and shapes in mathematics. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. The point where these axes intersect is called the origin.
Points in this plane are represented as (x, y), where x is the position on the horizontal axis and y is the position on the vertical axis. For the trigonometric context, the coordinate plane is instrumental in determining the trigonometric functions of an angle formed by the vertex at the origin.

• The x-value corresponds to the horizontal displacement from the origin.
• The y-value corresponds to the vertical displacement from the origin.
• The hypotenuse, or radius, is calculated as \( r = \sqrt{x^2 + y^2} \).

These elements together help define the sine, cosine, and tangent functions based on the coordinates of the point on the terminal side of the angle.

Reciprocal Functions

Trigonometric functions have corresponding reciprocal functions. These functions provide additional relationships and are equally significant in solving trigonometric problems. They stem from the basic trigonometric functions by simply taking the reciprocal.
Understanding reciprocal functions is essential as they are often required in more complex mathematical calculations. Here are the reciprocal relationships:

• The cosecant function, \( \csc\theta \), is the reciprocal of sine, \( \sin\theta \). It is given by \( \csc\theta = \frac{1}{\sin\theta} \).
• The secant function, \( \sec\theta \), is the reciprocal of cosine, \( \cos\theta \). It is given by \( \sec\theta = \frac{1}{\cos\theta} \).
• The cotangent function, \( \cot\theta \), is the reciprocal of tangent, \( \tan\theta \). It is given by \( \cot\theta = \frac{1}{\tan\theta} \).

While basic trigonometric functions describe ratios of the sides of a right triangle, their reciprocals provide a different perspective, enhancing problem-solving versatility.