Problem 43 - Algebra and Trigonometry

Question

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \sec \theta, \quad \tan \theta ; \quad \theta \text { in Quadrant II } $$

Step-by-Step Solution

Answer

\( \sec \theta = -\sqrt{1 + \tan^2 \theta} \)

Step 1: Define Relations in the Second Quadrant

In the second quadrant, the cosine of an angle is negative, and since \( \sec \theta = \frac{1}{\cos \theta} \), the secant will also be negative. The tangent \( \tan \theta \) is negative in the second quadrant because it is the ratio of sine to cosine.

Step 2: Use the Identity Relating Secant and Tangent

We use the Pythagorean identity: \( 1 + \tan^2 \theta = \sec^2 \theta \). This will help us express \( \sec \theta \) in terms of \( \tan \theta \).

Step 3: Solve the Identity for Secant

Rearranging the identity gives \( \sec^2 \theta = 1 + \tan^2 \theta \). Taking the square root of both sides gives \( \sec \theta = -\sqrt{1 + \tan^2 \theta} \) because \( \sec \theta \) is negative in the second quadrant.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a fundamental part of trigonometry. It is defined as the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \).
The role of the secant function becomes especially interesting when investigating its properties in different quadrants of the unit circle.

• In the first quadrant, both cosine and secant functions are positive.
• In the second quadrant, cosine is negative, making secant negative as well.
• In the third quadrant, cosine returns to being positive.
• Finally, in the fourth quadrant, cosine is positive, resulting in a positive secant value.

When dealing with trigonometric identities, the secant function frequently appears in conjunction with other trigonometric functions, such as sine and tangent, due to the interconnected nature of these identities.

Tangent Function

The tangent function, symbolized by \( \tan \theta \), is defined by the formula \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship stems from the sine and cosine functions and their respective values on the unit circle.

• In the first quadrant, tangent is positive because both sine and cosine are positive.
• In the second quadrant, tangent becomes negative since sine is positive and cosine is negative.
• The third quadrant sees tangent as positive again, as both sine and cosine are negative.
• In the fourth quadrant, tangent is negative because sine is negative while cosine is positive.

Tangent is particularly significant when exploring angles and their properties within the unit circle. Its importance is further highlighted when using identities such as the Pythagorean identity, \( 1 + \tan^2 \theta = \sec^2 \theta \).
This shows the close relationship between tangent and secant, providing a means to express one function in terms of the other.

Second Quadrant Properties

In trigonometry, each quadrant in the unit circle possesses unique properties due to the sign changes of the sine, cosine, and tangent functions. The second quadrant, which covers angles between 90° and 180°, has specific characteristics:

• Sine is positive: During the second quadrant, the y-coordinate of points on the unit circle, corresponding to sine, remains positive.
• Cosine is negative: The x-coordinate, representing cosine, turns negative, affecting functions like secant, which also become negative.
• Tangent is negative: As tangent is the ratio \( \frac{\sin \theta}{\cos \theta} \), a positive sine combined with a negative cosine results in a negative tangent.

Understanding these sign changes is crucial for solving problems that involve trigonometric identities, especially when determining the sign of functions based on their quadrant location. In our specific example, knowing that secant and tangent are negative in the second quadrant guides how we manipulate and solve the given trigonometric identity \( 1 + \tan^2 \theta = \sec^2 \theta \).
Recognizing these quadrant properties is invaluable for correctly simplifying expressions and solving equations in trigonometry.