Problem 71 - Algebra and Trigonometry

Question

Addition Formula for Tangent Use the Addition Formulas for Cosine and Sine to prove the Addition Formula for Tangent. [Hint: Use $$ \tan (s+t)=\frac{\sin (s+t)}{\cos (s+t)} $$ and divide the numerator and denominator by \(\cos s \cos t ]\)

Step-by-Step Solution

Answer

The addition formula for tangent is \( \tan(s+t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} \).

Step 1: State the Addition Formulas for Sine and Cosine

The addition formulas for sine and cosine are:- \( \sin(s+t) = \sin s \cos t + \cos s \sin t \)- \( \cos(s+t) = \cos s \cos t - \sin s \sin t \)These formulas will be utilized to express \( \tan(s+t) \) in terms of sine and cosine.

Step 2: Express Tangent Addition Formula

We start with the formula for tangent: \( \tan(s+t)= \frac{\sin(s+t)}{\cos(s+t)} \). We will express the sine and cosine of \(s+t\) in terms of \(s\) and \(t\).

Step 3: Substitute Sine and Cosine Addition Formulas

Substitute the formulas for sine and cosine:\[ \frac{\sin s \cos t + \cos s \sin t}{\cos s \cos t - \sin s \sin t} \]

Step 4: Divide Numerator and Denominator by \(\cos s \cos t\)

To simplify, we'll divide both the numerator and the denominator by \(\cos s \cos t\):\[ \frac{\frac{\sin s \cos t}{\cos s \cos t} + \frac{\cos s \sin t}{\cos s \cos t}}{\frac{\cos s \cos t}{\cos s \cos t} - \frac{\sin s \sin t}{\cos s \cos t}} \]

Step 5: Simplify Terms

Simplify each term:- \( \frac{\sin s}{\cos s} = \tan s \)- \( \frac{\sin t}{\cos t} = \tan t \)- \( \frac{\cos t}{\cos t} = 1 \)Substituting these into our equation, we have:\[ \frac{\tan s + \tan t}{1 - \tan s \tan t} \]

Step 6: State the Tangent Addition Formula

Thus, we have proven that the addition formula for tangent is:\[ \tan(s+t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} \]

Key Concepts

Addition Formula for Tangent

The addition formula for tangent is a fascinating relation in trigonometry that helps compute the tangent of the sum of two angles. This formula states that:\[ \tan(s+t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} \]The core idea behind this formula is to express the tangent function, which is the ratio of sine and cosine, so we start with:\[ \tan(s+t) = \frac{\sin(s+t)}{\cos(s+t)} \]Here the hint to solve it effectively is by dividing both the numerator and the denominator by \( \cos s \cos t \).

• First, express \( \sin(s+t) \) and \( \cos(s+t) \) using the addition formulas for sine and cosine.
• Then, substitute these expressions into the formula for tangent.
• Finally, simplifying by dividing through by \( \cos s \cos t \) results in the clear representation of the tangent addition formula.

This technique not only simplifies your calculations but also gives you a deeper insight into how trigonometric functions interrelate.

Addition Formulas for Sine and Cosine

The addition formulas for sine and cosine are essential tools in trigonometry. These formulas allow us to find the sine or the cosine of a sum of two angles, which are fundamental when working with complex expressions. The formulas are:

• Sine Addition Formula: \( \sin(s+t) = \sin s \cos t + \cos s \sin t \)
• Cosine Addition Formula: \( \cos(s+t) = \cos s \cos t - \sin s \sin t \)

To understand how these formulas work, consider the unit circle. Each angle on the unit circle corresponds to a point, represented by coordinates \( (\cos \theta, \sin \theta) \). When two angles are added, the resulting sine or cosine can be visualized as combining two such points on the unit circle.These addition formulas simplify expressions involving trigonometric functions and preserve the relationships between angles while incorporating elements of both sine and cosine.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions to simpler or more manageable forms. This is often essential in problem-solving to make calculations easier and results more comprehensible. A well-known approach involves identifying and applying trigonometric identities, such as the addition formulas or fundamental identities like \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).Steps in a typical simplification process:

• Identify suitable identities or formulas relevant to the expression at hand.
• Substitute these identities into the expression.
• Algebrically manipulate the resulting expression to reach a simpler form.

For example, simplifying \( \tan(s+t) \) using addition formulas involves breaking it down initially using sine and cosine, and then algebraic manipulation by dividing through by shared terms like \( \cos s \cos t \).This simplification significantly aids in solving trigonometric equations and proves invaluable in calculus, physics, and engineering applications where concise solutions are crucial.