Problem 10 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises 9 and \(10,\) find all the trigonometric values of \(\theta\) with the given conditions. $$\tan \theta=-1, \quad \sin \theta<0$$

Step-by-Step Solution

Answer

The solution to the problem is \( \theta = \frac{7\pi}{4} + 2n\pi \) or \( \theta = 315° + n.360° \) where n is an integer.

Step 1: Understand the given conditions

We are given two conditions: \( \tan \theta = -1 \) and \( \sin \theta < 0 \). The first implies that for some point on the unit circle, the y-coordinate divided by the x-coordinate is -1. The second implies that the y-coordinate of such point is negative which indicates the direction of the angle.

Step 2: Identify the Quadrants where the given conditions are met

Knowing that Tangent is negative in Q2 and Q4, and sine is negative in Q3 and Q4 of the unit circle. The meeting point for these conditions which \( \tan \theta = -1 \) and \( \sin \theta < 0 \) would be the 4th Quadrant (Q4).

Step 3: Find the Trigonometric Value of \( \theta \).

Since we have identified that the conditions meet in the 4th Quadrant (Q4), where \(\tan(\theta) = -1\), this happens at \(\theta = \frac{7\pi}{4}\) or \(\theta = 315°\) in the standard position where \(\theta\) rotates counterclockwise from the positive x-axis. The general solution will be \(\theta = 315° + n.360°\) or \(\theta = \frac{7\pi}{4} + 2n\pi \), where n is an integer.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Every point on the unit circle can be associated with an angle \( \theta \), measured from the positive x-axis.

The unit circle helps us understand trigonometric functions such as sine, cosine, and tangent. Here's how it works:

• The x-coordinate of a point on the unit circle represents \(\cos \theta\).
• The y-coordinate represents \(\sin \theta\).
• Tangent, or \( \tan \theta \), is the ratio of \(\sin \theta\) to \(\cos \theta\).

The circle is divided into four quadrants:

• Quadrant I: \(0^\circ\) to \(90^\circ\), where all trigonometric functions are positive.
• Quadrant II: \(90^\circ\) to \(180^\circ\), where sine is positive, and cosine and tangent are negative.
• Quadrant III: \(180^\circ\) to \(270^\circ\), where tangent is positive, and sine and cosine are negative.
• Quadrant IV: \(270^\circ\) to \(360^\circ\), where cosine is positive, and sine and tangent are negative.

Understanding the unit circle and its quadrants is essential for solving trigonometry problems like the given exercise.

Tangent Function

The tangent of an angle, \( \theta \), is a key trigonometric function. It is defined as the ratio of the sine of the angle to the cosine of the angle:\[\tan \theta = \frac{\sin \theta}{\cos \theta}.\]Because it is a ratio, the tangent function can tell us a lot about the relationships between angles and sides in triangles, especially right triangles.

In the context of the unit circle, the tangent function may also be seen as the slope of the line connecting the origin to the point \((\cos \theta, \sin \theta)\). This function is interesting because:

• It has a periodic nature and repeats every \(180^\circ\).
• It is positive when both sine and cosine have the same sign, which happens in the first and third quadrants.
• It is negative when sine and cosine have different signs, which happens in the second and fourth quadrants.

For the exercise, \( \tan \theta = -1 \) indicates that the y-coordinate \((\sin \theta)\) is the negative reciprocal of the x-coordinate \((\cos \theta)\), and the context of the unit circle tells us this occurs in the fourth quadrant when sine is negative.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. They are powerful tools for simplifying trigonometric expressions and solving equations. Some of the most important include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Tangent Identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• Reciprocal Identities, such as \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \)

These identities not only simplify expressions but also help solve trigonometric equations by creating relationships among the functions.

For instance, knowing that \( \tan \theta = -1 \) allows us to infer connections between sine and cosine that are governed by these identities. In the particular exercise, using these identities along with constraints \( \tan \theta = -1 \) and \( \sin \theta < 0 \) helps locate \( \theta \) precisely in the fourth quadrant.