Problem 28 - Algebra and Trigonometry

Question

find the exact value of each of the remaining trigonometric functions of \(\theta\) $$ \cos \theta=\frac{1}{3}, \quad 270^{\circ}<\theta<360^{\circ} $$

Step-by-Step Solution

Answer

Therefore, the values of the trigonometric functions are: \( \sin \theta = - \sqrt{8}/3 \), \( \tan \theta = - \sqrt{8} \), \( \csc \theta = - \sqrt{8}/2 \), \( \sec \theta = 3 \), and \( \cot \theta = - 1/\sqrt{8} \).

Step 1: Apply the Pythagorean Identity

Use the Pythagorean Identity to determine the value of sin(\(\theta\)). Start with the equation \(\sin^2(\theta) + \cos^2(\theta) = 1\). Substitute 1/3 for cos(\(\theta\)) and solve for sin(\(\theta\)). This gives two possible solutions so just select the one that is negative (since sin is negative in the 4th quadrant).

Step 2: Evaluate the Other Trigonometric Functions

Once sin(\(\theta\)) is known, calculate the values for tan(\(\theta\)), csc(\(\theta\)), sec(\(\theta\)), and cot(\(\theta\)) using the equations tan(\(\theta\)) = sin(\(\theta\))/cos(\(\theta\)), csc(\(\theta\)) = 1/sin(\(\theta\)), sec(\(\theta\)) = 1/cos(\(\theta\)), and cot(\(\theta\)) = cos(\(\theta\))/sin(\(\theta\)). Again, determine the appropriate signs for each value based on the quadrant of \(\theta\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry. It relates the squares of the sine and cosine functions of an angle. Mathematically, it is expressed as: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \]This identity is derived from the Pythagorean Theorem and the concept of the unit circle.
To find an unknown trigonometric function when another is known, substitute the known value into this identity and solve for the unknown. In the exercise, we know \( \cos \theta = \frac{1}{3} \). Substituting this into the Pythagorean Identity lets us solve for \( \sin \theta \).

• Start by squaring \( \cos \theta \): \( \left(\frac{1}{3}\right)^2 = \frac{1}{9} \).
• Substitute it into the identity: \( \sin^2(\theta) + \frac{1}{9} = 1 \).
• Rearrange to find \( \sin^2(\theta) = 1 - \frac{1}{9} = \frac{8}{9} \).
• Solve for \( \sin(\theta) \) by taking the square root: \( \sin(\theta) = \pm \sqrt{\frac{8}{9}} = \pm \frac{2\sqrt{2}}{3} \).

The correct sign must be chosen based on the quadrant in which the angle lies.

Trigonometric Identities

Trigonometric identities are equations that hold true for all angles. They are a crucial part of solving problems in trigonometry as they provide relationships between the trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.
Once \( \sin(\theta) \) is found using the Pythagorean Identity, the other trigonometric functions can be determined using these identities:

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

In the exercise, after finding \( \sin(\theta) = -\frac{2\sqrt{2}}{3} \), you can plug this into the identities to find the exact values of the other trigonometric functions.
Determining the correct sign for each function is vital and depends on the angle's quadrant.

Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. A unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.
On the unit circle, the x-coordinate represents \( \cos(\theta) \) and the y-coordinate represents \( \sin(\theta) \). The angle \( \theta \) is measured from the positive x-axis in a counter-clockwise direction.

• The circle provides a straightforward way to visualize and comprehend how trigonometric functions change as the angle \( \theta \) varies.
• All trigonometric functions can be derived from points on this circle based on the definitions involving the x and y coordinates.

In our exercise, the knowledge that \( \cos(\theta) = \frac{1}{3} \) and \( 270^\circ < \theta < 360^\circ \) tells us that the angle is in the fourth quadrant of the unit circle. This influences the sign of trigonometric values, like making \( \sin(\theta) \) negative.

Fourth Quadrant Behavior

Understanding the behavior of trigonometric functions in different quadrants is essential for solving trigonometry problems.
The fourth quadrant is where angles between \( 270^\circ \) and \( 360^\circ \) lie. The signs of trigonometric functions in this quadrant are peculiar:

• \( \cos(\theta) \) is positive.
• \( \sin(\theta) \) is negative.
• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) is also negative because a negative divided by a positive is negative.

Knowing which functions are positive or negative in each quadrant helps correctly resolve exercises dealing with trigonometric functions. In our scenario, this means choosing the negative solution for \( \sin(\theta) \) to ensure the values correspond with the angle's position relative to the coordinate axes, aiding in accurate calculation of other trigonometric identities.