Problem 42 - Precalculus Mathematics for Calculus

Question

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$\sec \theta, \quad \sin \theta ; \quad \theta \text { in Quadrant } 1$$

Step-by-Step Solution

Answer

\( \sec \theta = \frac{1}{\sqrt{1 - \sin^2 \theta}} \) in Quadrant 1.

Step 1: Understand the Relationship Between Functions

In trigonometry, the secant function is defined as the reciprocal of the cosine function, i.e., \( \sec \theta = \frac{1}{\cos \theta} \). Meanwhile, the sine function is directly related to the cosine function through the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).

Step 2: Solve for Cosine in Terms of Sine

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \cos \theta \) in terms of \( \sin \theta \):\[ \cos^2 \theta = 1 - \sin^2 \theta \]\[ \cos \theta = \sqrt{1 - \sin^2 \theta} \]Since \( \theta \) is in Quadrant 1, both sine and cosine are positive, so we use the positive square root.

Step 3: Express Secant in Terms of Sine

Now substitute \( \cos \theta \) from Step 2 into the definition of the secant:\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\sqrt{1 - \sin^2 \theta}} \]

Step 4: Final Expression

Thus, the expression for \( \sec \theta \) in terms of \( \sin \theta \) is:\[ \sec \theta = \frac{1}{\sqrt{1 - \sin^2 \theta}} \].

Key Concepts

Secant and Sine Relationship

In trigonometry, understanding the relationships between different functions can be very helpful. This is particularly true for secant and sine. Secant (\( \sec \theta \)) is the reciprocal of cosine, meaning it is defined as \( \sec \theta = \frac{1}{\cos \theta} \). This shows how tightly secant is linked to cosine.

Sine (\( \sin \theta \)) is another foundational function. The relationship between sine and cosine is expressed through the trigonometric identity\( \sin^2 \theta + \cos^2 \theta = 1 \). This equation not only connects sine and cosine but also helps us express one in terms of the other.

Whenever you're asked to express secant in terms of sine, you'll need to use this identity to replace cosine with a function of sine. This step is a crucial bridge for transforming secant into an expression involving only sine.

Trigonometric Identities

Trigonometric identities are like the backbone of trigonometry. They are equations that relate the trigonometric functions to one another. One of the most important identities, often called the Pythagorean identity, is:

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

This identity is derived from the Pythagorean theorem and is fundamental in trigonometry. It allows us to find one trigonometric function in terms of another.

For instance, if you know the value of sine, you can find cosine by rearranging the identity to:\( \cos^2 \theta = 1 - \sin^2 \theta \). By taking the square root, you can solve for cosine and consequently for secant.

Since identities hold true for any angle \( \theta \), they are powerful tools in solving trigonometric problems, making it easier to switch between different functions.

Quadrant Analysis

Trigonometry divides the coordinate plane into four quadrants, each affecting the sign of the trigonometric functions. Understanding the behavior of functions in these quadrants is key to solving trigonometry problems accurately.

In Quadrant 1, where \( \theta \) is considered, both sine and cosine are positive. This is why when we solve \( \cos \theta \) as \( \sqrt{1 - \sin^2 \theta} \), we select the positive square root. Knowing the signs of the functions is essential for ensuring the accuracy of your solution.

So, when working with trigonometric functions, it's important to:

• Determine in which quadrant the angle \( \theta \) resides.
• Know the trigonometric function signs in that quadrant.

This quadrant analysis helps in applying the correct trigonometric transformations consistently and accurately, making sure that the calculated values reflect the correct mathematical behavior.