Problem 9 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises 9 and \(10,\) find all the trigonometric values of \(\theta\) with the given conditions. $$\cos \theta=-\frac{15}{17}, \quad \sin \theta>0$$

Step-by-Step Solution

Answer

The trigonometric values of \(\theta\) are: \(\sin \theta = \frac{8}{17}\), \(\cos \theta = -\frac{15}{17}\), \(\tan \theta = -\frac{8}{15}\), \(\csc \theta = \frac{17}{8}\), \(\sec \theta = -\frac{17}{15}\), and \(\cot \theta = -\frac{15}{8}\).

Step 1: Identifying the quadrant

\(\cos \theta\) is negative and \(\sin \theta\) is positive, so we know that \(\theta\) is in the second quadrant.

Step 2: Finding the sine value

Now we need to find the \(\sin \theta\). We know that \(\cos^2 \theta + \sin^2 \theta = 1\). Thus, \(\sin \theta = \sqrt{1 - \cos^2 \theta}\). We substitute \(\cos \theta = -\frac{15}{17}\) and we find that \(\sin \theta = \frac{8}{17}\) (we use positive here because we know that \(\sin \theta > 0\) since we're in the second quadrant).

Step 3: Getting other trigonometric values

We now calculate the remaining trigonometric values based on \(\sin \theta\) and \(\cos \theta\): \(\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{8}{15}\), \(\csc \theta = \frac{1}{\sin \theta} = \frac{17}{8}\), \(\sec \theta = \frac{1}{\cos \theta} = -\frac{17}{15}\), \(\cot \theta = \frac{1}{\tan \theta} = -\frac{15}{8}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools that help us relate different trigonometric functions to each other. One of the most fundamental identities is the Pythagorean identity. This identity states that for any angle \( \theta \), the square of the sine of \( \theta \) plus the square of the cosine of \( \theta \) equals one. Mathematically, it is written as:
\[\cos^2 \theta + \sin^2 \theta = 1\]
This identity is really useful because it allows us to find one trigonometric function if we know the other. For example, if we know \( \cos \theta \) and want to find \( \sin \theta \), we can rearrange the identity to:
\[\sin \theta = \sqrt{1 - \cos^2 \theta}\]

• This is particularly helpful when combined with information about the sign, such as \( \sin \theta > 0 \).
• It's important to use correct signs when taking square roots, depending on the quadrant.

Quadrants in Trigonometry

Understanding which quadrant an angle is in is key to determining the signs of trigonometric functions. The coordinate plane is divided into four quadrants:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, and cosine is negative.
• Third Quadrant: Sine is negative, and cosine is negative.
• Fourth Quadrant: Sine is negative, and cosine is positive.

In this exercise, we determined that \( \cos \theta = -\frac{15}{17} \) and \( \sin \theta > 0 \). This indicates the angle \( \theta \) is in the second quadrant, since that's where cosine is negative and sine is positive.
Recognizing the correct quadrant allows us to correctly choose the signs for the calculated values of sine and other trigonometric functions.

Sine and Cosine Relationships

Sine and cosine have a close relationship expressed through their properties and identities, which are especially clear when considering angles in the different quadrants. For instance, in the unit circle, cosine represents the x-coordinate and sine represents the y-coordinate. This spatial relationship helps predict their behaviors and values:

• The Pythagorean identity involves both sine and cosine, linking their squares.
• The identity \( \sin^2 \theta + \cos^2 \theta = 1 \) shows how they complement each other.

In the problem, with \( \cos \theta = -\frac{15}{17} \), finding \( \sin \theta \) involves confirming its sign according to the quadrant. With \( \sin \theta \) found to be \( \frac{8}{17} \), the relationship between sine and cosine guides us to find tangent, cosecant, secant, and cotangent:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{8}{15} \)
• \( \csc \theta = \frac{1}{\sin \theta} = \frac{17}{8} \)
• \( \sec \theta = \frac{1}{\cos \theta} = -\frac{17}{15} \)
• \( \cot \theta = \frac{1}{\tan \theta} = -\frac{15}{8} \)

These co-function relationships provide a comprehensive understanding of trigonometric functions beyond just sine and cosine.