Problem 39 - Precalculus: Functions and Graphs

Question

Exer. \(37-46:\) Verify the identity. $$\tan \left(u+\frac{\pi}{4}\right)=\frac{1+\tan u}{1-\tan u}$$

Step-by-Step Solution

Answer

The identity is verified; both sides of the equation match.

Step 1: Recall the Tangent Addition Formula

The formula for the tangent of the sum of two angles is given by:\[\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}.\]In this problem, we have \(a = u\) and \(b = \frac{\pi}{4}\).

Step 2: Use the Tangent of \(\frac{\pi}{4}\)

Recall that \(\tan\left(\frac{\pi}{4}\right) = 1\). Substitute \(\tan(b) = 1\) into the tangent addition formula:\[\tan(u + \frac{\pi}{4}) = \frac{\tan(u) + 1}{1 - \tan(u) \cdot 1}.\]

Step 3: Simplify the Expression

Simplify the equation:\[\tan(u + \frac{\pi}{4}) = \frac{\tan(u) + 1}{1 - \tan(u)}.\]This matches the identity we want to verify.

Key Concepts

Tangent Addition Formula

The Tangent Addition Formula is a crucial identity in trigonometry that helps us find the tangent of the sum of two angles. It allows us to express \( \tan(a + b) \) in terms of the tangents of the individual angles \( a \) and \( b \). The general formula is:\[ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \]When using this formula, it is important to remember the following:

• The presence of complementary signs in the denominator ensures the accurate calculation of tangent values over different quadrants.
• This formula is particularly useful when dealing with angles that aren't directly available on the unit circle.

In simple terms, the Tangent Addition Formula helps us in handling the sum or difference of angles in trigonometric equations, making complex expressions more manageable.

Angle Sum Identities

Angle Sum Identities are formulas that provide the trigonometric function values of the sum or difference of two angles. These identities are essential in simplifying expressions that involve multiple angles.There are three primary Angle Sum Identities:

• For Sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• For Cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• For Tangent: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \)

These identities are extremely beneficial when working with problems involving angles that aren't straightforward. By converting angles into sums or differences, these identities simplify the process of finding exact trigonometric values.
This conversion is especially helpful when dealing with half angles, double angles, or angles that exceed 90 degrees.

Trigonometric Functions

Trigonometric Functions are fundamental in understanding angles and their relationships in mathematics. The primary trigonometric functions are Sine \( \sin \), Cosine \( \cos \), and Tangent \( \tan \). Each function links a given angle to a ratio derived from a right triangle:

• Sine (\( \sin \theta \)): Represents the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): Represents the ratio of the opposite side to the adjacent side.

These functions form the basis for more complex trigonometric identities and equations. They are pivotal in various applications such as physics, engineering, and even music theory. Understanding how these functions interact is key to solving trigonometric problems. By using identities like the Tangent Addition Formula, learners can navigate complicated trigonometric scenarios confidently and effectively.