Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-14,\) for each given function value, find the remaining five trigonometric function values. \(\sec \theta=-8\) and \(\theta\) is in the second quadrant.

Step-by-Step Solution

Answer

The six trigonometric functions are: \( \sin \theta = \frac{\sqrt{63}}{8} \), \( \cos \theta = \frac{-1}{8} \), \( \tan \theta = -\sqrt{63} \), \( \sec \theta = -8 \), \( \csc \theta = \frac{8}{\sqrt{63}} \), \( \cot \theta = -\frac{1}{\sqrt{63}} \).

Step 1: Understand the Relationship

The secant function, \( \sec \theta \), is defined as the reciprocal of the cosine function. Thus, \( \sec \theta = \frac{1}{\cos \theta} \). Given \( \sec \theta = -8 \), then \( \cos \theta = \frac{-1}{8} \).

Step 2: Recognize the Quadrant Characteristics

Since \( \theta \) is in the second quadrant, cosine is negative, which matches our calculation. Additionally, sine is positive in the second quadrant.

Step 3: Find Sine Using Pythagorean Identity

The Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) allows us to solve for \( \sin \theta \). Substitute \( \cos \theta = \frac{-1}{8} \) into this identity: \[\sin^2 \theta + \left(\frac{-1}{8}\right)^2 = 1\]\[\sin^2 \theta + \frac{1}{64} = 1\]\[\sin^2 \theta = 1 - \frac{1}{64} = \frac{63}{64}\]\[\sin \theta = \sqrt{\frac{63}{64}} = \frac{\sqrt{63}}{8}\].Since sine is positive in the second quadrant, \( \sin \theta = \frac{\sqrt{63}}{8} \).

Step 4: Calculate Remaining Trigonometric Functions

Now compute the other trigonometric functions:- \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{63}}{8}}{\frac{-1}{8}} = -\sqrt{63}\)- \( \csc \theta = \frac{1}{\sin \theta} = \frac{8}{\sqrt{63}}\)- \( \cot \theta = \frac{1}{\tan \theta} = -\frac{1}{\sqrt{63}}\).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a crucial trigonometric concept. It is defined as the reciprocal of the cosine function. This means:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Understanding this relationship is vital, especially when given \( \sec \theta = -8 \). From this, we can derive that \( \cos \theta = \frac{-1}{8} \).

The negative value of the secant signifies that the cosine of the angle \( \theta \) is also negative. This typically occurs in the second and third quadrants. Identifying the sign of these functions is essential for determining the angle's appropriate position.

When dealing with trigonometric functions, recognizing the reciprocal relationships helps simplify calculations and ensure that assumptions about the angle's properties are accurate.

Pythagorean Identity

The Pythagorean identity is foundational in trigonometry. It is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity links sine and cosine, providing a way to calculate one when the other is known. Given \( \cos \theta = \frac{-1}{8} \), we use this identity to find \( \sin \theta \).

Substituting into the identity, we have:\[\sin^2 \theta + \left(\frac{-1}{8}\right)^2 = 1\]\[\sin^2 \theta + \frac{1}{64} = 1\]\[\sin^2 \theta = 1 - \frac{1}{64} = \frac{63}{64}\]The solution for \( \sin \theta \) is \( \frac{\sqrt{63}}{8} \), noting that in the second quadrant, sine is positive.

Mastering how to manipulate this identity allows seamless transition between sine and cosine values, which is helpful when finding other trigonometrical values.

Second Quadrant

The location of an angle \( \theta \) in the coordinate plane dictates the signs of its trigonometric functions. In the second quadrant:

• Cosine and secant are negative: \( \sec \theta \), owing to \( \cos \theta \)'s negativity.
• Sine and cosecant are positive, as seen with \( \sin \theta = \frac{\sqrt{63}}{8} \).

Being in the second quadrant, we expect the tangent \( \tan \theta \) to also be negative because \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This highlights one of trigonometry's critical aspects: understanding angles' positions and how they affect trigonometric signs and values.

Thus, when dealing with trigonometric functions in different quadrants, always recall the specific behaviors for sine, cosine, and tangent.