Problem 13 - Algebra 2

Question

Find the value of each expression. \(\tan \theta,\) if \(\sec \theta=-3 ; 180^{\circ}<\theta<270^{\circ}\)

Step-by-Step Solution

Answer

\(\tan \theta = 2\sqrt{2}\).

Step 1: Understand the given quadrant

The interval \(180^{\circ} < \theta < 270^{\circ}\) tells us that \(\theta\) is in the third quadrant. In the third quadrant, both sine and cosine are negative, and thus, tangent is positive.

Step 2: Relate \(\sec \theta\) to \(\cos \theta\)

Since \(\sec \theta = \frac{1}{\cos \theta}\), we know that \(\sec \theta = -3\). Therefore, \(\cos \theta = \frac{-1}{3}\).

Step 3: Use Pythagorean identity

Recall the identity \(\sec^2 \theta = 1 + \tan^2 \theta\). Substitute \(\sec \theta = -3\) into the identity: \((-3)^2 = 1 + \tan^2 \theta\).

Step 4: Solve for \(\tan \theta\)

Now simplify the equation: \(9 = 1 + \tan^2 \theta\). Subtract 1 from both sides to get \(8 = \tan^2 \theta\). Take the square root of both sides: \(\tan \theta = \pm 2\sqrt{2}\).

Step 5: Determine the sign of \(\tan \theta\)

Since \(\theta\) is in the third quadrant, where tangent is positive, we have \(\tan \theta = 2\sqrt{2}\).

Key Concepts

Tangent Function

The tangent function, noted as \( \tan \theta \), is one of the fundamental functions in trigonometry. It is defined as the ratio of the sine to the cosine of an angle \( \theta \):

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Tangent provides us with the slope of the line that the angle \( \theta \) makes with the x-axis when drawn in standard position. This function is especially useful when determining angles where the definition of sine and cosine alone does not suffice.

In trigonometric problem solving, tangent is frequently used to determine unknown ratios or angles. Its value can be positive or negative depending on the quadrant in which the angle lies. Thus, understanding the behavior of the tangent function is critical in interpreting angles in various geometric contexts.

Pythagorean Identity

The Pythagorean Identities are foundational in trigonometry. They relate the squares of trigonometric functions to 1, which serves as a great tool for simplifying and solving trigonometric equations. One such identity is:

• \( \sec^2 \theta = 1 + \tan^2 \theta \).

This equation is derived from the Pythagorean Theorem, linking the secant function and the tangent function. It allows us to solve for one trigonometric function in terms of the others.

In our original step-by-step solution, this identity plays a crucial role in finding the value of \( \tan \theta \). By knowing the secant value, we can easily solve for tangent using this identity. Rearranging and substituting into the identity enables a deeper understanding of the relationships between the trigonometric ratios.

Trigonometric Quadrants

Understanding trigonometric quadrants is essential when dealing with angles beyond the first quadrant (0 to 90 degrees). The coordinate plane is divided into four quadrants, each determining the sign of the trigonometric functions:

• Quadrant I: All functions (sine, cosine, tangent) are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

In our exercise, \( \theta \) is situated in the third quadrant, where the tangent function is positive while both sine and cosine are negative. This characteristic guides us in determining the correct sign of \( \tan \theta \), ensuring an accurate solution when applying trigonometric identities.

Secant Function

The secant function, noted as \( \sec \theta \), is one of the six primary trigonometric functions and is closely related to the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \).

Secant represents the reciprocal of cosine, providing a measure that's part of the trigonometric ratios in a triangle.

In the context of the exercise, knowing that \( \sec \theta = -3 \) implies that \( \cos \theta = \frac{-1}{3} \). This negative value indicates that the angle \( \theta \) is in one of the quadrants where cosine is negative, such as the second or third quadrant. Understanding the nature of the secant function helps to bridge the concept from cosine, aiding in comprehending and solving trigonometric equations more effectively.