Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

\( \csc \theta = \frac{1}{\sqrt{1 - \cos^2 \theta}} \)

Step 1: Understand the Relationship between Trigonometric Functions

The cosecant function is defined as the reciprocal of the sine function. Therefore, we know that \( \csc \theta = \frac{1}{\sin \theta} \). The cosine function \( \cos \theta \) is related to sine by the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).

Step 2: Express Sine in Terms of Cosine

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \sin \theta \) in terms of \( \cos \theta \):\[ \sin^2 \theta = 1 - \cos^2 \theta \]Then take the square root of both sides, considering only the positive root since \( \theta \) is acute:\[ \sin \theta = \sqrt{1 - \cos^2 \theta} \]

Step 3: Write Cosecant in Terms of Cosine

Substitute the expression for \( \sin \theta \) into the definition of cosecant:\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\sqrt{1 - \cos^2 \theta}} \]

Step 4: Finalize the Expression

Therefore, the expression for \( \csc \theta \) in terms of \( \cos \theta \) is:\[ \csc \theta = \frac{1}{\sqrt{1 - \cos^2 \theta}} \]

Key Concepts

Cosecant Function

The extbf{cosecant function} is one of the six fundamental trigonometric functions. It is symbolized as \( \csc \theta \) and serves as the reciprocal function of the sine.

• If you know the value of sine for an angle, you can easily find the cosecant by taking its reciprocal.
• This relationship is expressed mathematically as \( \csc \theta = \frac{1}{\sin \theta} \).

This definition highlights that the cosecant function is undefined where \( \sin \theta = 0 \), since division by zero is not possible. For example, this typically occurs when \( \theta = 0 \) or \( \theta = 180^\circ \).
In practical applications, the cosecant function is often less intuitive to understand than sine or cosine, as it deals with longer hypotenuses compared to opposite sides in right-angled triangles.
However, knowing it is simply the reciprocal of sine helps make it more accessible in solving trigonometry problems.

Cosine Function

The extbf{cosine function}, represented as \( \cos \theta \), is another primary trigonometric function. It is intimately linked to the geometry of right-angled triangles.

• The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse.
• In the context of the unit circle, \( \cos \theta \) corresponds to the x-coordinate of a point on the circle's circumference.

The unit circle helps illustrate how the cosine function behaves as an angle increases, remaining between -1 and 1 for all values of \( \theta \).
Understanding cosine is crucial in problems involving oscillations, waves, and circular motions. Its symmetrical properties also help simplify more complex analyses in fields like physics and engineering.
Knowing its relationship with sine through identities offers powerful tools for transforming expressions, like using the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \).

Pythagorean Identity

The extbf{Pythagorean identity} is a central piece in trigonometry, drawing its name from the famous Pythagorean theorem. The identity connects the sine and cosine functions in a straightforward equation:

• The standard form is \( \sin^2 \theta + \cos^2 \theta = 1 \).
• This equation applies to any angle \( \theta \), meaning it's universally applicable across all trigonometric scenarios.

What makes this identity so powerful is the ability to solve equations and transform parts of a trigonometric function into another. For instance, by rearranging terms:
- To find \( \sin \theta \) in terms of \( \cos \theta \), use \( \sin^2 \theta = 1 - \cos^2 \theta \).
- Conversely, \( \cos^2 \theta = 1 - \sin^2 \theta \) allows you to express cosine in terms of sine.

The identity's role extends beyond mere problem-solving. It aids in the visualization of trigonometric functions on the unit circle, where every point represents \( (\cos \theta, \sin \theta) \). Understanding the Pythagorean identity is essential not only to grasp basic trigonometric identities but also to explore further mathematical fields such as calculus and analytical geometry.