Problem 17 - Algebra 2 and Trigonometry

Question

In \(11-18, P\) is a point on the terminal side of an angle in standard position with measure \(\theta\) and on a circle with center at the origin and radius \(r .\) For each point \(P,\) find: a. \(r\) b. \(\csc \theta\) c. \(\sec \theta\) d. \(\cot \theta\) \((9,-9)\)

Step-by-Step Solution

Answer

r = 9√2, csc θ = -√2, sec θ = √2, cot θ = -1

Step 1: Find r

\(r = \sqrt{x^2 + y^2} = \sqrt{9^2 + (-9)^2} = \sqrt{81+81} = \sqrt{162} = 9\sqrt{2}\)

Step 2: Find csc θ

\(\sin\theta = \frac{y}{r} = \frac{-9}{9\sqrt{2}} = \frac{-1}{\sqrt{2}}\)

\(\csc\theta = \frac{1}{\sin\theta} = -\sqrt{2}\)

Step 3: Find sec θ

\(\cos\theta = \frac{x}{r} = \frac{9}{9\sqrt{2}} = \frac{1}{\sqrt{2}}\)

\(\sec\theta = \frac{1}{\cos\theta} = \sqrt{2}\)

Step 4: Find cot θ

\(\tan\theta = \frac{y}{x} = \frac{-9}{9} = -1\)

\(\cot\theta = \frac{1}{\tan\theta} = -1\)

Key Concepts

Standard Position Angle

A standard position angle is a fundamental concept in trigonometry that helps define the position of an angle on a coordinate plane. It is drawn from the origin, where the initial side lies along the positive x-axis. The terminal side then rotates counterclockwise to reach its final position.
Understanding the standard position is crucial:

• It provides a uniform way to describe angles.
• Makes it easier to connect trigonometry to geometry and algebra.
• Helps in identifying the quadrant in which the terminal side lies.

For example, in the exercise, the point \(9, -9\) determines the terminal side of the angle, leading us into the third quadrant based on its coordinates. Recognizing this can guide us in finding various trigonometric functions with respect to the angle \(\theta\).

Cosecant Function

The cosecant function, denoted as \(\csc \theta\), is the reciprocal of the sine function. It can be expressed as:\[\csc \theta = \frac{1}{\sin \theta}\]
To compute \(\csc \theta\), you'll first need to understand the sine of angle \(\theta\) in standard position:

• Sine represents the ratio of the opposite side over the hypotenuse.
• \(\csc \theta\) will thus be the ratio of the hypotenuse over the opposite side.

In a coordinate system, the opposite side is represented by the \(y\)-value of the point on the terminal side. Thus, given point \(P(9, -9)\) and the radius \(r = 9\sqrt{2}\), we determine:\[\csc \theta = \frac{9\sqrt{2}}{-9} = -\sqrt{2}\] which provides the reciprocal of the sine function for the angle \(\theta\).

Secant Function

The secant function, represented as \(\sec \theta\), is the reciprocal of the cosine function. It is defined mathematically by the ratio:\[\sec \theta = \frac{1}{\cos \theta}\]
Here’s how you compute \(\sec \theta\) using coordinates:

• Cosine is the adjacent side over the hypotenuse.
• For \(\sec \theta\), it becomes the hypotenuse over the adjacent side.
• In the coordinate plane, this translates to \(\frac{r}{x}\).

Given the point \(9, -9)\) and \(r = 9\sqrt{2}\), the calculation becomes:\[\sec \theta = \frac{9\sqrt{2}}{9} = \sqrt{2}\]
This result translates how \(\sec \theta\) can be used in the analysis of the angle \(\theta\) in its standard position.

Cotangent Function

The cotangent function \(\cot \theta\) is the reciprocal of the tangent function. Its formula is:\[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]
When dealing with points, you can also determine \(\cot \theta\) through:

• The ratio of the adjacent to the opposite side.
• Which in the coordinate context translates to \(\frac{x}{y}\).

Using the point \(9, -9\), the calculation is straightforward:\[\cot \theta = \frac{9}{-9} = -1\]
This reveals \(\cot \theta\) as another important function for angles in standard position, giving further insight into the triangle formed by the radius \(r\).