Problem 25 - Precalculus Essentials

Question

In Exercises \(23-34\), find the exact value of each of the remaining trigonometric functions of \(\theta .\) $$\sin \theta=\frac{5}{13}, \quad \theta \text { in quadrant II }$$

Step-by-Step Solution

Answer

The exact values of the trigonometric functions of \( \theta \) are \( \sin \theta = \frac{5}{13}, \cos \theta = - \frac{12}{13}, \tan \theta = - \frac{5}{12}, \csc \theta = \frac{13}{5}, \sec \theta = - \frac{13}{12}, \cot \theta = - \frac{12}{5} \).

Step 1: Determine the value of \( \cos \theta \) using Pythagorean Identity

The first step is to calculate the value of \( \cos \theta \) using the Pythagorean Identity \( \cos^2 \theta = 1 - \sin^2 \theta \). Substituting the given value of \( \sin \theta = \frac{5}{13} \), we obtain \( \cos^2 \theta = 1 – \left(\frac{5}{13} \right)^2 = 1 – \frac{25}{169} = \frac{144}{169} \). Therefore, \( \cos \theta = \pm \sqrt{\frac{144}{169}} = \pm \frac{12}{13} \). However, since \( \theta \) is in the second quadrant and the cosine function is negative in the second quadrant, we conclude that \( \cos \theta = - \frac{12}{13} \).

Step 2: Calculate the trigonometric functions secant, tangent, and cotangent

Now, we can calculate the value of the remaining trigonometric functions. The secant is the reciprocal of the cosine, so we get \( \sec \theta = - \frac{13}{12} \). Using the relation \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we find \( \tan \theta = - \frac{5}{12} \). The cotangent is the reciprocal of the tangent, which gives us \( \cot \theta = - \frac{12}{5} \).

Step 3: Calculate the trigonometric function cosecant

The cosecant function is the reciprocal of the sine, so \( \csc \theta = \frac{13}{5} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that establishes a relationship between the sine and cosine of an angle. It's derived from the Pythagorean Theorem, which states that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In trigonometric terms, this identity can be expressed as:

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

Using the Pythagorean Identity, you can easily find one trigonometric function if you already know the other. For example, if you know that \( \sin \theta = \frac{5}{13} \), you can substitute this value into the identity and solve for \( \cos \theta \).
This involves some algebraic manipulation:

• First, compute \( \sin^2 \theta = \left(\frac{5}{13}\right)^2 = \frac{25}{169} \).
• Then, substituting into the identity gives: \( 1 = \frac{25}{169} + \cos^2 \theta \),
• Solving for \( \cos^2 \theta \), you get \( \cos^2 \theta = \frac{144}{169}. \)

This result allows you to find \( \cos \theta \), bearing in mind the sign of the function depends on the angle's quadrant.

Sine and Cosine Relationship

Sine and cosine are two of the primary trigonometric functions, and they are closely related both mathematically and geometrically. These functions are especially significant because they allow us to express angles and their related dimensions in circular and wave-like forms.
In many scenarios, you can use the relationships between sine and cosine to find missing information. For example, once you know \( \sin \theta \), the cos of the same angle can be found using the Pythagorean Identity as previously discussed, which interrelates them through the equation:

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

When solving problems, remember that both values work together to place an angle in its correct position in a circle, reflecting its geometric nature.
Moreover, the relationship between sine and cosine can also be understood through their graphs, which essentially are phase-shifted versions of each other. This graphical insight helps in visualizing how these functions increase and decrease in values depending on the angle.

Quadrant Analysis

Quadrant analysis is all about understanding where angles and their trigonometric functions fall on the unit circle. Each quadrant on the unit circle has unique characteristics that affect the sign (positive or negative) of trigonometric functions.
The unit circle is divided into four quadrants:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, but cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, but cosine is positive.

In our exercise, the given \( \sin \theta = \frac{5}{13} \) places \( \theta \) in Quadrant II. Here, the sine function retains a positive value while the cosine turns negative. This knowledge is crucial, especially when determining the correct sign for \( \cos \theta \), as stated in the provided step-by-step solution. Understanding quadrant analysis is essential for correcting calculate other trigonometric functions like tangent and secant without making sign errors.