Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Use fundamental identities to write the first expression in terms of the second, for any acute angle \(\theta\). $$\sec \theta, \sin \theta$$

Step-by-Step Solution

Answer

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

Step 1: Recall the Definition of Secant

The secant of an angle \( \theta \) is defined as \( \sec \theta = \frac{1}{\cos \theta} \). This is a reciprocal identity in trigonometry.

Step 2: Use the Pythagorean Identity

The Pythagorean identity states that \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity will help us express \( \cos \theta \) in terms of \( \sin \theta \).

Step 3: Solve for \( \cos \theta \)

From the Pythagorean identity, we can express \( \cos \theta \) as \( \cos \theta = \sqrt{1 - \sin^2 \theta} \). We use the positive root because the angle \( \theta \) is acute.

Step 4: Substitute in the Secant Definition

Now that we have \( \cos \theta = \sqrt{1 - \sin^2 \theta} \), we can substitute this into the secant formula: \( \sec \theta = \frac{1}{\sqrt{1 - \sin^2 \theta}} \).

Step 5: Final Expression

Thus, \( \sec \theta \) in terms of \( \sin \theta \) is \( \sec \theta = \frac{1}{\sqrt{1 - \sin^2 \theta}} \). This concludes the transformation of \( \sec \theta \) into an expression using \( \sin \theta \).

Key Concepts

Secant Function

The secant function is a trigonometric function that is closely related to the cosine function. It is defined in terms of the cosine function and is one of the reciprocal trigonometric functions. The secant of an angle \( \theta \) is written as \( \sec \theta \), and it is defined by the following expression:

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

This definition shows that the secant function is the reciprocal of the cosine function.
When \( \cos \theta \) is known, the secant function can be easily calculated, provided that \( \cos \theta \) is not zero because division by zero is undefined. The secant function helps in many trigonometric transformations and problem solving, as seen when expressing other functions like sine in terms of secant using fundamental identities. This reciprocal relationship is vital in converting expressions or solving equations in trigonometry.

Sine Function

The sine function is one of the basic functions in trigonometry and is a fundamental concept for understanding angles and triangles. Usually denoted by \( \sin \theta \), it is defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

• For a given angle \( \theta \), \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

The sine function is important not only in geometry but it also plays a significant role in various formulas and identities in trigonometry, which are essential for solving complex problems.
The value of \( \sin \theta \) always lies between -1 and 1, and it is used extensively in conjunction with the cosine and tangent functions to describe periodic phenomena like sound and light waves. Additionally, transformations involving the sine function are crucial, such as expressing the secant function in terms of the sine function, which aids in simplifying trigonometric expressions.

Pythagorean Identity

The Pythagorean identity is one of the key trigonometric identities that forms the foundation of many problems and transformations within trigonometry. This identity expresses a fundamental relationship between the sine and cosine functions:

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

This identity is derived from the Pythagorean Theorem, which relates the sides of a right-angled triangle.
The Pythagorean identity is useful because it allows us to express either \( \sin \theta \) in terms of \( \cos \theta \), or \( \cos \theta \) in terms of \( \sin \theta \). For example, if we know the value of \( \sin \theta \), we can find \( \cos \theta = \sqrt{1 - \sin^2 \theta} \).
This conversion is particularly helpful when solving equations that involve trigonometric functions or when converting different trigonometric expressions.
The ability to replace one trigonometric function with another using the Pythagorean identity simplifies many complex problems and helps in deriving other important trigonometric identities.