Problem 8 - Precalculus

Question

The terminal side of an angle \(\theta\) in standard position passes through values of the six trigonometric functions for angle \(\theta\) $$(-9,-5)$$

Step-by-Step Solution

Answer

The six trigonometric functions are: \( \sin \theta = \frac{-5}{\sqrt{106}}, \cos \theta = \frac{-9}{\sqrt{106}}, \tan \theta = \frac{5}{9}, \csc \theta = \frac{\sqrt{106}}{-5}, \sec \theta = \frac{\sqrt{106}}{-9}, \cot \theta = \frac{9}{5} \).

Step 1: Define the hypotenuse

To find the trigonometric values for angle \( \theta \), first identify the hypotenuse of the triangle formed by the x-coordinate and y-coordinate of the point \((-9,-5)\). This requires calculating the distance from the origin to this point using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-9)^2 + (-5)^2} = \sqrt{81 + 25} = \sqrt{106} \]. Thus, the hypotenuse \( r = \sqrt{106} \).

Step 2: Compute Sine and Cosine

Now, compute \(\sin \theta\) and \(\cos \theta\) using the point's y and x values along with the hypotenuse. The sine function is defined as: \( \sin \theta = \frac{y}{r} = \frac{-5}{\sqrt{106}} \). The cosine function is defined as: \( \cos \theta = \frac{x}{r} = \frac{-9}{\sqrt{106}} \).

Step 3: Compute Tangent

The tangent of \(\theta\) is the ratio of the sine to the cosine, or \( \tan \theta = \frac{y}{x} = \frac{-5}{-9} = \frac{5}{9} \).

Step 4: Compute Cosecant, Secant, and Cotangent

The cosecant is the reciprocal of the sine: \( \csc \theta = \frac{r}{y} = \frac{\sqrt{106}}{-5} \). The secant is the reciprocal of the cosine: \( \sec \theta = \frac{r}{x} = \frac{\sqrt{106}}{-9} \). The cotangent is the reciprocal of the tangent: \( \cot \theta = \frac{x}{y} = \frac{-9}{-5} = \frac{9}{5} \).

Key Concepts

Angle Theta

In trigonometry, angles are often denoted by theta, represented by the Greek letter \( \theta \). Understanding the angle \( \theta \) is fundamental when learning about trigonometric functions. The angle \( \theta \) is not just a number; it defines the rotation or orientation of a line in a coordinate plane.

When dealing with an angle \( \theta \), you look at where the terminal side ends up, which is an infinite ray starting from the origin (0,0) and moving through a given point such as (-9, -5).

• Angles can be measured in degrees or radians; both are essential in different contexts.
• Understanding how angles affect the positions and lengths in triangles is key to mastering trigonometric functions.

Learning about \( \theta \) involves comprehending its application in real-world scenarios and mathematical problems, often serving as the stepping stone to more complex trigonometric concepts.

Standard Position

The term "standard position" in trigonometry refers to an angle positioned on the coordinate plane. Specifically, an angle is in standard position when:

• Its vertex is located at the origin \((0,0)\) of the coordinate plane.
• The initial side lies along the positive x-axis.

When we say that the angle \( \theta \) is in standard position and its terminal side passes through the point \((-9, -5)\), it means that if you draw a line from the origin to this point, the angle this line makes with the positive x-axis is \( \theta \).

This concept is fundamental in defining the trigonometric functions because it allows for consistent measurement of angles and helps in visualizing how the sine, cosine, and other trigonometric functions relate to their respective sides on the coordinate plan.

Reciprocal Identities

Reciprocal identities are essential in trigonometry because they expand how we understand and compute trigonometric functions. Let's look at the six main trigonometric functions and their reciprocal identities:

• The **sine**, \( \sin \theta \), has a reciprocal called the **cosecant**, \( \csc \theta = \frac{1}{\sin \theta} \). In our exercise, this translates to \( \csc \theta = \frac{\sqrt{106}}{-5} \).
• The **cosine**, \( \cos \theta \), has a reciprocal known as the **secant**, \( \sec \theta = \frac{1}{\cos \theta} \). Here, \( \sec \theta = \frac{\sqrt{106}}{-9} \).
• The **tangent**, \( \tan \theta \), is reciprocated by the **cotangent**, \( \cot \theta = \frac{1}{\tan \theta} \). In our specific case, \( \cot \theta = \frac{9}{5} \).

The reciprocal identities are notable not just for calculation purposes, but because they offer different perspectives on how to view and understand angles and their relationships within triangles. These identities are invaluable within proofs and help simplify complex trigonometric expressions.