Problem 44 - Precalculus: Functions and Graphs

Question

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\tan \theta, \cos \theta$$

Step-by-Step Solution

Answer

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

Step 1: Recall Identity Linking Tan and Sin, Cos

The tangent of an angle can be expressed in terms of sine and cosine using the fundamental identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). We will use this identity to express \( \tan \theta \) in terms of \( \cos \theta \).

Step 2: Use the Pythagorean Identity for Sine

To express \( \sin \theta \) in terms of \( \cos \theta \), we use the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Solving for \( \sin \theta \), we have \( \sin \theta = \sqrt{1 - \cos^2 \theta} \).

Step 3: Substitute in the Tangent Identity

Substitute \( \sin \theta = \sqrt{1 - \cos^2 \theta} \) into \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This yields: \( \tan \theta = \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} \).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a crucial trigonometric function frequently used in mathematics, particularly in trigonometry. The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. Mathematically, it can be expressed as:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]This identity highlights how the tangent function is directly related to both the sine and cosine functions.
The tangent function is especially important when dealing with right triangles, as it represents the ratio of the opposite side to the adjacent side of a given angle.
By understanding this function, students can better grasp relationships in geometric applications. Since the tangent function deals with sine and cosine, it connects various trigonometric concepts, making it a valuable tool for solving a wide range of mathematical problems.

Pythagorean Identity

The Pythagorean Identity is one of the fundamental identities in trigonometry and plays a vital role in relating the sine and cosine functions. It states that for any angle \( \theta \), the following equation holds:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is derived from the Pythagorean theorem and provides a powerful method for expressing one trigonometric function in terms of another.
For instance, if we know \( \cos \theta \), we can find \( \sin \theta \) by rearranging the identity:

• Solve for \( \sin^2 \theta \) by subtracting \( \cos^2 \theta \) from 1: \( \sin^2 \theta = 1 - \cos^2 \theta \).
• To find \( \sin \theta \), take the square root: \( \sin \theta = \sqrt{1 - \cos^2 \theta} \).

Understanding the Pythagorean Identity allows students to navigate between sine and cosine, giving them flexibility in their problem-solving approach.
This identity is particularly useful in exercises where one trigonometric function needs to be expressed in terms of another, as seen in the original exercise.

Sine and Cosine Relationship

The sine and cosine functions are fundamental in trigonometry, each describing a specific aspect of right-angled triangles and circular functions. Their relationship is expressed elegantly through multiple identities, the most notable being the tangent function and the Pythagorean Identity.Both functions define the coordinates on the unit circle:

• \( \sin \theta \) represents the y-coordinate.
• \( \cos \theta \) represents the x-coordinate.

Together, they establish the basis for many trigonometric identities and equations.
The relationship between sine and cosine in the tangent function, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), expresses how they can be used interchangeably to solve problems when one component is known.
Understanding their inherent relationship allows students to manipulate and transform trigonometric expressions easily.
This foundational knowledge is crucial when exploring more advanced concepts in mathematics or physics.
Overall, grasping the sine and cosine relationship is key to mastering trigonometric identities and functions.