Problem 40 - Algebra 2

Question

Express the first trigonometric function in terms of the second. $$ \tan \theta, \cos \theta $$

Step-by-Step Solution

Answer

The first trigonometric function \( \tan \theta \) can be expressed in terms of the second \( \cos \theta \) as \( \tan \theta = \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} \).

Step 1: Write \( \tan \theta \) using sin and cos

The trigonometric identity of \( \tan \theta \) is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Step 2: Express \( \sin \theta \) in terms of \( \cos \theta \)

The Pythagorean identity \( \sin^2 \theta + \cos^2 \theta =1 \) can be rearranged to solve for \( \sin \theta \) as \( \sin \theta= \sqrt{1 - \cos^2 \theta} \)

Step 3: Substitute \( \sin \theta \) in the equation

Step 1 and 2 are substituted into the original equation of \( \tan \theta \) in terms of \( \sin \theta \) and \( \cos \theta \) yielding \( \tan \theta = \frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are key equations that involve trigonometric functions, such as sine, cosine, and tangent. They are fundamental tools for simplifying trigonometric expressions and proving other mathematical statements. One of the most frequently encountered identities is the identity for tangent:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This identity allows us to express the tangent function, \( \tan \theta \), in terms of sine and cosine functions. By understanding how these identities relate the trigonometric functions, you can manipulate and combine expressions in different useful ways. Trigonometric identities form the basis for more complex identities and help in solving a wide range of trigonometric equations.

Pythagorean Identity

The Pythagorean identity is an essential trigonometric identity that relates the square of the sine and cosine functions. It is named for its resemblance to the Pythagorean theorem in geometry. The Pythagorean identity is given as:

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

This identity is particularly useful for finding one function in terms of another. For example, if you know \( \cos \theta \), you can find \( \sin \theta \) using the rearranged formula:

• \( \sin \theta = \sqrt{1 - \cos^2 \theta} \)

This rearrangement is achieved by isolating \( \sin^2 \theta \) and taking the square root, which is a critical step in many trigonometric problems. By applying this identity, you can derive expressions involving other trigonometric functions, making it an indispensable tool for expressing and simplifying trigonometric relationships.

Trigonometric Expressions

Trigonometric expressions are combinations of trigonometric functions that represent angles in various mathematical problems. These expressions can often be simplified or re-expressed using known trigonometric identities. For instance, to express \( \tan \theta \) in terms of \( \cos \theta \), you begin with the identity:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

To replace \( \sin \theta \) with a term involving only \( \cos \theta \), use the Pythagorean identity:

• \( \sin \theta = \sqrt{1 - \cos^2 \theta} \)

By substituting \( \sin \theta \) into the initial expression for \( \tan \theta \), you arrive at:

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

Expressions like this are vital in calculus, physics, and engineering, where expressing one trigonometric function in terms of another can simplify complex equations and allow for more straightforward calculations.