Problem 33 - Algebra and Trigonometry

Question

find the exact value of each of the remaining trigonometric functions of \(\theta\) $$ \sec \theta=-3, \quad \tan \theta>0 $$

Step-by-Step Solution

Answer

The exact values of each of the remaining trigonometric functions of \(\theta\) are: \n cos(\(\theta\)) = -1/3, \n sin(\(\theta\)) = -2/3, \n tan(\(\theta\)) = 2, \n csc(\(\theta\)) = -3/2, \n cot(\(\theta\)) = 1/2

Step 1: Find the quadrant where the angle lies

From the given information, one can infer that sec(\(\theta\)) is negative and tan(\(\theta\)) is positive. Consequently, \(\theta\) lies within the third quadrant because sec, which is reciprocal of cosine, is negative in the second and third quadrants, whereas tan, is positive in the first and third quadrants.

Step 2: Determine cosine

The secant is the reciprocal of the cosine, therefore, to find cos(\(\theta\)), the reciprocal of \(-3\) is taken, which is \(-1/3\).

Step 3: Determine sine using Pythagorean identity

The Pythagorean identity states that \(sin^2\theta + cos^2\theta = 1\). We already know that \(cos\theta = -1/3\). Then, the sine can be found by rearranging and solving the equation. \(sin\theta = ±\sqrt{1 - cos^2\theta} = ±\sqrt{1 - (-1/3)^2} = ±\sqrt{8/9} = ±2/3\). Since we determined \(\theta\) is in the third quadrant, \(sin(\theta)\) will be negative. Thus, \(sin(\theta) = -2/3\)

Step 4: Establish the other trigonometric functions

Now that we have sin(\(\theta\)) and cos(\(\theta\)), the other trigonometric function values can be calculated as follows: \n- tan(\(\theta\)) = sin(\(\theta\)) / cos(\(\theta\)) = (-2/3) / (-1/3) = 2 \n- cot(\(\theta\)) = 1/ tan ( \(\theta\)) = 1/2 \n- csc(\(\theta\)) = 1/ sin(\(\theta\)) = -3/2

Key Concepts

Quadrant Identification

To find the exact value of trigonometric functions, it's essential to first determine the quadrant in which an angle lies. This is done by understanding the signs of trigonometric functions:

• Secant (\( \sec \theta \)) is the reciprocal of cosine, and if \( \sec \theta \) is negative, \( \theta \) could be in either the second or third quadrant.
• Tangent (\( \tan \theta \)) is positive, meaning \( \theta \) must be in the first or third quadrant.

Combining both conditions, \( \theta \) must be in the third quadrant. In this quadrant, cosine and secant are negative, and tangent is positive. By identifying which quadrant the angle resides in, we can more easily determine the signs of the trigonometric functions.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry connecting sine and cosine:
\[\sin^2\theta + \cos^2\theta = 1\]
Knowing this identity allows us to find one trigonometric function if another is known. In the exercise, we start with \( \cos \theta = -\frac{1}{3} \). Using the Pythagorean identity:
\[\sin \theta = \pm \sqrt{1 - \cos^2\theta} = \pm \sqrt{1 - \left(-\frac{1}{3}\right)^2} = \pm \sqrt{\frac{8}{9}}\]
This simplifies to \( \pm \frac{2}{3} \). Since \( \theta \) is in the third quadrant where sine is negative, \( \sin \theta = -\frac{2}{3} \). The Pythagorean identity helps us deduce relationships and find exact values.

Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals. They are crucial for solving exercises that involve these functions. Here are some key reciprocal identities:

• Secant is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \)
• Cosecant is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \)
• Cotangent is the reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \)

For \( \sec \theta = -3 \), we find \( \cos \theta = -\frac{1}{3} \). Once we have \( \sin \theta = -\frac{2}{3} \), the reciprocal identity gives us \( \csc \theta = -\frac{3}{2} \). Knowing the relations between functions and their reciprocals simplifies finding the remaining function values.