Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\log \tan \theta=\log \sin \theta-\log \cos \theta$$

Step-by-Step Solution

Answer

The identity is verified: \( \log \tan \theta = \log \sin \theta - \log \cos \theta \).

Step 1: Understand the Identity

We are asked to verify the identity \( \log \tan \theta = \log \sin \theta - \log \cos \theta \). We need to transform the lefthand side (\( \log \tan \theta \)) into the righthand side (\( \log \sin \theta - \log \cos \theta \)).

Step 2: Recall Properties of Logarithms

The property \( \log \left( \frac{a}{b} \right) = \log a - \log b \) will be particularly useful for transforming the left side into the right side of the identity.

Step 3: Express Tangent in terms of Sine and Cosine

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute this expression into the log function: \( \log \tan \theta = \log \left( \frac{\sin \theta}{\cos \theta} \right) \).

Step 4: Apply Logarithmic Property

Use the property from Step 2: \( \log \left( \frac{\sin \theta}{\cos \theta} \right) = \log \sin \theta - \log \cos \theta \).

Step 5: Verify the Identity

Now we see that \( \log \tan \theta = \log \sin \theta - \log \cos \theta \), which transforms the left side into the right side, confirming that the identity holds.

Key Concepts

Logarithms

Logarithms are an important mathematical concept used to solve exponential equations. A logarithm answers the question: to what power must a certain base be raised, to obtain a given number. For example, in the equation \( abla_x = y \), \( x \) is the exponent and y is the result of the base \( abla \) raised to that exponent. The statement \( \log_b(y) = x \) expresses this. Logarithms transform multiplication into addition and division into subtraction.
Key properties of logarithms include:

• \( \log(ab) = \log a + \log b \)
• \( \log \left(\frac{a}{b}\right) = \log a - \log b \)
• \( \log a^b = b \cdot \log a \)

These properties allow complex multiplications or divisions to be calculated more easily. In the given exercise, we utilized the property \( \log \left( \frac{a}{b} \right) = \log a - \log b \) to verify the identity.

Tangent Function

The tangent function is one of the primary functions in trigonometry, alongside sine and cosine. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. More simply, it can be expressed in terms of sine and cosine. The formula for tangent is:

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

This relationship is crucial in trigonometry. For the given exercise, this formula was used to express \( \tan \theta \) in terms of sine and cosine. By substituting \( \frac{\sin \theta}{\cos \theta} \) into the logarithm, we were able to transform and verify the identity. Tangent function plays a significant role in oscillations, waves, and circular motion.
The tangent function periodically repeats and is undefined for angles where the cosine is zero, making it essential to understand its limits and applications.

Sine Function

The sine function is another fundamental function in trigonometry. It relates the angle in a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. Mathematically, it is expressed as:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

Sine values oscillate between -1 and 1. This function is crucial for modeling periodic phenomena like sound and light waves. In the identity exercise, \( \sin \theta \) is a part of the expression \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Understanding how sine functions alongside cosine can simplify many trigonometric identities is important for mastering math problems. Sine is also a part of Euler's formula, which connects complex numbers with trigonometric functions. This makes the sine function indispensable in both theory and application.

Cosine Function

The cosine function is closely associated with the sine function in trigonometry. It relates the angle in a right-angled triangle to the ratio of the length of the adjacent side to the hypotenuse. The formula for cosine is:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Much like the sine, cosine values also oscillate between -1 and 1. The cosine function describes how much of a quantity goes along the "x" axis in a unit circle, whereas sine describes the "y" axis. In the identity \( \log \tan \theta = \log \sin \theta - \log \cos \theta \), \( \cos \theta \) appears in the denominator of the tangent function expression. Recognizing how cosine interacts with other trigonometric functions helps simplify complex identities.
The cosine function is fundamental in many fields including physics, engineering, and computer graphics, often used to model cycles or periodic phenomena.