Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-14,\) for each given function value, find the remaining five trigonometric function values. \(\cos \theta=-\frac{3}{4}\) and \(\theta\) is in the third quadrant.

Step-by-Step Solution

Answer

The remaining trigonometric functions are: \( \sin \theta = -\frac{\sqrt{7}}{4} \), \( \tan \theta = \frac{\sqrt{7}}{3} \), \( \sec \theta = -\frac{4}{3} \), \( \cot \theta = \frac{3}{\sqrt{7}} \), \( \csc \theta = -\frac{4}{\sqrt{7}} \).

Step 1: Identify Quadrant and Reference Angles

Since \( \theta \) is in the third quadrant, both sine and cosine are negative. We will use the given \( \cos \theta = -\frac{3}{4} \) to find the other values. We need to remember that in the third quadrant, the tangent is positive, and the reference angle will have the same sine and cosine values but both negative.

Step 2: Use Pythagorean Identity

The Pythagorean identity tells us that \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = -\frac{3}{4} \) into the identity and solve for \( \sin \theta \):\[ \sin^2 \theta + \left(-\frac{3}{4}\right)^2 = 1 \]\[ \sin^2 \theta + \frac{9}{16} = 1 \]\[ \sin^2 \theta = 1 - \frac{9}{16} \]\[ \sin^2 \theta = \frac{16}{16} - \frac{9}{16} = \frac{7}{16} \]So, \( \sin \theta = -\sqrt{\frac{7}{16}} = -\frac{\sqrt{7}}{4} \) because sine is negative in the third quadrant.

Step 3: Find Tangent

Tangent is sine divided by cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \):\[ \tan \theta = \frac{-\frac{\sqrt{7}}{4}}{-\frac{3}{4}} = \frac{\sqrt{7}}{3} \]

Step 4: Find Cotangent

Cotangent is the reciprocal of tangent, \( \cot \theta = \frac{1}{\tan \theta} \):\[ \cot \theta = \frac{1}{\frac{\sqrt{7}}{3}} = \frac{3}{\sqrt{7}} \].

Step 5: Find Secant

Secant is the reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \):\[ \sec \theta = \frac{1}{-\frac{3}{4}} = -\frac{4}{3} \].

Step 6: Find Cosecant

Cosecant is the reciprocal of sine, \( \csc \theta = \frac{1}{\sin \theta} \):\[ \csc \theta = \frac{1}{-\frac{\sqrt{7}}{4}} = -\frac{4}{\sqrt{7}} \].

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental relation in trigonometry. It is used to connect the sine and cosine values of an angle in the following equation: \[ \sin^2 \theta + \cos^2 \theta = 1 \]This equation allows us to find one trigonometric function value when given another. If you know the value of either sine or cosine, you can substitute it into this identity to solve for the other. For instance, in our exercise, we are given \( \cos \theta = -\frac{3}{4} \). Substituting this into the identity helps us find \( \sin^2 \theta \) as: \[ \sin^2 \theta + \left(-\frac{3}{4}\right)^2 = 1 \]By solving this, we can determine \( \sin \theta = -\frac{\sqrt{7}}{4} \), keeping in mind the sign change as per the conditions of the angle's quadrant. This identity is a crucial tool for students solving trigonometric problems.

third quadrant

In trigonometry, understanding which quadrant an angle lies in helps us determine the signs of its trigonometric function values. The coordinate plane is divided into four quadrants. Each is marked by different signs for sine, cosine, and tangent values:

• Quadrant I: \( \sin \theta > 0, \cos \theta > 0, \tan \theta > 0 \)
• Quadrant II: \( \sin \theta > 0, \cos \theta < 0, \tan \theta < 0 \)
• Quadrant III: \( \sin \theta < 0, \cos \theta < 0, \tan \theta > 0 \)
• Quadrant IV: \( \sin \theta < 0, \cos \theta > 0, \tan \theta < 0 \)

Our exercise places \( \theta \) in the third quadrant. Here, both sine and cosine values are negative while the tangent is positive. Recognizing these sign characteristics is vital in accurately finding the other trigonometric function values.

sine and cosine values

The sine and cosine of an angle are foundational concepts in trigonometry. Sine, denoted by \( \sin \theta \), represents the y-coordinate on the unit circle, while cosine, denoted by \( \cos \theta \), represents the x-coordinate. In our exercise, we were given \( \cos \theta = -\frac{3}{4} \) in the third quadrant, meaning both are negative. In general:

• Sine is positive in the first two quadrants and negative in the third and fourth.
• Cosine is positive in the first and fourth quadrants and negative in the second and third.

After using the Pythagorean identity, we determined \( \sin \theta = -\frac{\sqrt{7}}{4} \). This negative value reflects the quadrant's influence. Understanding these values helps in calculating other trigonometric functions like tangent, which is derived from \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

reciprocal identities

Reciprocal identities are key relationships in trigonometry that translate one trigonometric function into another by taking their reciprocal. These identities are commonly used to find secant, cosecant, and cotangent from cosine, sine, and tangent:

• Secant: \( \sec \theta = \frac{1}{\cos \theta} \)
• Cosecant: \( \csc \theta = \frac{1}{\sin \theta} \)
• Cotangent: \( \cot \theta = \frac{1}{\tan \theta} \)

In the given problem, calculating these identities is straightforward once we have \( \sin \theta \) and \( \cos \theta \):

• \( \sec \theta = -\frac{4}{3} \)
• \( \csc \theta = -\frac{4}{\sqrt{7}} \)
• \( \cot \theta = \frac{3}{\sqrt{7}} \)

These reciprocal functions, alongside the primary functions, provide a comprehensive way to express and interrelate the six crucial trigonometric functions of any angle.