Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\csc \theta=-\frac{3}{2}, \tan \theta < 0$$

Step-by-Step Solution

Answer

The six trigonometric functions of the angle are: \(\sin \theta = -\frac{2}{3}\), \(\cos \theta = -\frac{\sqrt{5}}{3}\), \(\tan \theta = \frac{2\sqrt{5}}{5}\), \(\csc \theta = -\frac{3}{2}\), \(\sec \theta =-\frac{3\sqrt{5}}{5}\) and \(\cot \theta = \frac{\sqrt{5}}{2}\).

Step 1: Conversion

Convert the given cosecant value into sine value. As \(\csc \theta\) and \(\sin \theta\) are reciprocal to each other, \(\sin \theta = \frac{1}{\csc \theta} =\frac{1}{-\frac{3}{2}}=-\frac{2}{3}\).

Step 2: Find Cosine Value

Make use of the Pythagorean identity, \(\sin^2 \theta+\cos^2 \theta=1\), to compute \(\cos \theta\). Substituting the value of \(\sin \theta\) we find that \(\cos \theta= ±\sqrt{1-\sin^2 \theta}= ±\sqrt{1-(-\frac{2}{3})^2}=±\frac{\sqrt{5}}{3}\).

Step 3: Determine the sign of Cosine Value

Determine the sign of the cosine function based on the given information that \(\tan \theta\) is negative. This tells us that the angle lies in either the second quadrant or the fourth quadrant, where the cosine function is negative and positive respectively. However, as \(\sin \theta\) is negative, the angle lies in the fourth quadrant (where both sine and tangent are negative), hence \(\cos \theta = -\frac{\sqrt{5}}{3}\).

Step 4: Find Remaining Trigonometric Functions

Compute the remaining trigonometric functions using the values of \(\sin \theta\) and \(\cos \theta\). \(\tan \theta = \frac{\sin \theta}{\cos \theta}= \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}}= \frac{2}{\sqrt{5}}= \frac{2\sqrt{5}}{5}\); \(\sec \theta = \frac{1}{\cos \theta} =-\frac{3\sqrt{5}}{5}\); \(\cot \theta = \frac{1}{\tan \theta} = \frac{\sqrt{5}}{2}\).

Key Concepts

Cosecant Function

The cosecant function is one of the six fundamental trigonometric functions. It is symbolized as \( \csc \theta \), which stands for cosecant of the angle \( \theta \). In any right triangle, it is the ratio of the length of the hypotenuse to the length of the opposite side relative to the angle. Simply put:

• \( \csc \theta = \frac{1}{\sin \theta} \)

To convert from cosecant to sine, as shown in the exercise, you take the reciprocal of the cosecant value. For example, if \( \csc \theta = -\frac{3}{2} \), then \( \sin \theta = -\frac{2}{3} \). This conversion is crucial, as evaluating sine can help us apply additional identities to find other trigonometric values.
Understanding this reciprocal relationship is often the first step in solving problems involving the cosecant function. By transforming the function into its more common counterpart, you can navigate through complex trigonometric expressions more easily.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry. It relates the squares of sine and cosine functions to 1. The basic formulation is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is derived from the Pythagorean theorem in right triangles. Whenever you know one of the squares, you can find the other. For instance, given \( \sin \theta = -\frac{2}{3} \), you can plug this into the identity to find \( \cos \theta \):
\(\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(-\frac{2}{3}\right)^2 = \frac{5}{9}\)

Therefore, \( \cos \theta \) would be \( ±\frac{\sqrt{5}}{3} \).
Verifying signs depends on the quadrant analysis, shedding light on whether \( \cos \theta \) is positive or negative.
This identity is invaluable for solving and simplifying trigonometric expressions, especially when only partial information is provided.

Quadrant Analysis

Quadrant analysis helps determine the signs of trigonometric functions based on the angle's position on the Cartesian coordinate system. It divides the plane into four quadrants:

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive; cosine and tangent are negative.
• Third Quadrant: Tangent is positive; sine and cosine are negative.
• Fourth Quadrant: Cosine is positive; sine and tangent are negative.

Given \( \csc \theta = -\frac{3}{2} \) and \( \tan \theta < 0 \), this implies the angle lies in the fourth quadrant. In this quadrant, \( \sin \theta \) and \( \tan \theta \) are negative, while \( \cos \theta \) remains positive.
However, from the exercise, it turns out \( \cos \theta \) is also negative, reaffirming why it's vital always to use quadrant analysis to verify signs. It paints a clearer picture of angles' positional relevance, eliminating potential missteps in calculations.