Problem 24 - Algebra and Trigonometry

Question

find the exact value of each of the remaining trigonometric functions of \(\theta\) $$ \sin \theta=-\frac{12}{13}, \quad \theta \text { in quadrant III } $$

Step-by-Step Solution

Answer

The exact values of the remaining trigonometric functions are: cos θ = -5/13, tan θ = 12/5, sec θ = -13/5, csc θ = -13/12, cot θ = 5/12.

Step 1: Verify quadrant and values

We're told θ lies in the third quadrant. Also, sin θ = -12/13. This is really the y-coordinate of a point on the unit circle if we draw the angle in standard position. Because the angle lies in the 3rd quadrant, x will be negative as well.

Step 2: Find cos θ by Pythagorean identity

Since sin^2 θ + cos^2 θ = 1, we can rearrange to find cos^2 θ = 1 - sin^2 θ, which gives us cos^2 θ = 1 - (-12/13)^2 = 1 - 144/169 = 25/169. Remember we stated because we're in the third quadrant, cos is also negative. Thus, cos θ = -√(25/169) = -5/13.

Step 3: Calculate tan θ using ratio of sin to cos

By definition, tan θ = sin θ / cos θ. We plug in the values we found: tan θ = (-12/13) / (-5/13) = 12/5.

Step 4: Find sec θ, csc θ and cot θ by taking reciprocals

Recall that sec θ is 1/cos θ, csc θ is 1/sin θ, and cot θ is 1/tan θ. So, sec θ = -13/5, csc θ = -13/12, and cot θ = 5/12.

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is essential for anyone learning trigonometry. It is based on the Pythagorean theorem which relates the lengths of the sides in a right-angled triangle. The identity itself expresses a fundamental relationship between the sine and cosine of an angle, stating that for any angle \theta, the square of the sine plus the square of the cosine always equals one: \[ \sin^2(\theta) + \cos^2(\theta) = 1.\]
In the context of our exercise, when given the value of the sine function and the quadrant wherein the angle lies, the Pythagorean identity helps us deduce the cosine function's value. Since the angle is in the third quadrant, we know both sine and cosine values are negative. Applying the identity, we calculate the cosine's value and ensure we assign the appropriate sign based on the quadrant. This fundamental identity not only helps in solving for unknown trigonometric values but also reinforces the concept of sine and cosine as coordinates on the unit circle.

Unit Circle

The unit circle is a cornerstone concept of trigonometry and aids in visualizing and understanding trigonometric functions. It's a circle with a radius of one, centered at the origin of a coordinate plane. The points on the unit circle are at a distance of one from the origin, which simplifies calculations.
When we talk about the angle \(\theta\) in trigonometry, we often refer to the angle formed by the intersection of the terminal side of \(\theta\) with the unit circle and the positive x-axis. The x-coordinate of this point is the cosine of the angle, and the y-coordinate is the sine. Hence, in the exercise, the sine value of \(-\frac{12}{13}\) represents the y-coordinate of a point on the unit circle where the angle \(\theta\) terminates. Since the radius of the unit circle is always one, this leads to a simple and immediate application of the Pythagorean identity when working with trigonometric problems.

Quadrants of The Coordinate System

The coordinate system is divided into four regions or quadrants, each with unique characteristics for the signs of x (cosine) and y (sine) values. In the first quadrant, both x and y are positive; in the second, x is negative and y is positive; in the third quadrant, where our exercise takes place, both x and y are negative; and in the fourth quadrant, x is positive and y is negative.
Knowing the quadrant is crucial as it directly affects the sign of the trigonometric functions. For instance, a negative sine value already hints that the angle lies either in the third or fourth quadrant. Since our exercise specifies the angle is in the third quadrant, we determine the cosine must also be negative. This understanding is imperative when finding the exact trigonometric values for an angle and ensures that our solutions are consistent with the geometrical interpretation of trigonometric functions on the coordinate system.