Problem 55 - Algebra and Trigonometry

Question

Derive the identity for \(\tan (\alpha+\beta)\) using $$ \tan (\alpha+\beta)=\frac{\sin (\alpha+\beta)}{\cos (\alpha+\beta)} $$ After applying the formulas for sums of sines and cosines, divide the numerator and denominator by \(\cos \alpha \cos \beta\)

Step-by-Step Solution

Answer

The identity for \(\tan (\alpha+\beta)\) is \(\tan (\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\), provided \(\cos\alpha\cos\beta\neq0\).

Step 1: Applying Sum Formulas for Sine and Cosine

First, replace \(\sin(\alpha+\beta)\) and \(\cos(\alpha+\beta)\) by their sum formulas: \(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\) and \(\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\). The resulting formula is \(\tan (\alpha+\beta)=\frac{\sin\alpha\cos\beta+\cos\alpha\sin\beta}{\cos\alpha\cos\beta-\sin\alpha\sin\beta}\).

Step 2: Divide Numerator and Denominator by \(\cos \alpha \cos \beta\)

Next, divide both the numerator and denominator of the previous fraction by \(\cos \alpha \cos \beta\), which results in \(\tan (\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\), where \(\tan\alpha = \frac{\sin\alpha}{\cos\alpha}\) and \(\tan\beta = \frac{\sin\beta}{\cos\beta}\).

Step 3: Summarize the Result

Finally, we have found the desired identity \(\tan (\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\), which holds as long as \(\cos\alpha\cos\beta\neq0\), that is, neither of \(\alpha\) and \(\beta\) is an odd multiple of \(\frac{\pi}{2}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for any value of the variable where the functions are defined. Think of them as the trigonometry toolbelt that helps us simplify complex expressions or solve trigonometric equations. They include the Pythagorean identities, reciprocal identities, and, importantly for this discussion, the sum and difference formulas. These formulas express the sine, cosine, or tangent of the sum or difference of two angles in terms of the sines and cosines of the individual angles.

For instance, the sum formula for sine, \(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\), allows us to find the sine of a sum of two angles using only the sine and cosine of each angle independently. This formula, along with its counterparts for cosine and tangent, is not only essential for solving complex trigonometric problems but is the backbone in deriving other trigonometric identities like the one for \(\tan(\alpha + \beta)\).

Sum Formulas for Sine and Cosine

The sum formulas for sine and cosine are pivotal in trigonometry, especially when we're dealing with angles that are not readily found on the unit circle. The sum formula for sine is \(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\), and for cosine, it is \(\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta\).

These formulas enable us to break down more complex angles into simpler components that we are more familiar with. An intuitive way to understand why we need these formulas is to imagine having to find the sine or cosine of a 75-degree angle. You probably won't find that angle on any unit circle diagram, but using the sum formulas, you could break this down using 45 degrees and 30 degrees, which are familiar angles.

Tangent Function

The tangent function, denoted as \(\tan\), is one of the six fundamental trigonometric functions. In a right triangle, it's defined as the ratio of the opposite side to the adjacent side. In the unit circle context, it's the ratio of the sine to the cosine of an angle. An intriguing aspect of the tangent function is that it repeats every \(\pi\) radians (or 180 degrees), known as its period, and has asymptotes where the cosine of an angle is zero. This is because the tangent function can be seen as sine divided by cosine, \(\tan \alpha = \frac{\sin\alpha}{\cos\alpha}\).

Understanding the tangent function is essential when using the tangent sum identity, as it involves both addition and division of tangent values. The derived formula, \(\tan (\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\), suggests a remarkable relationship that adding angles within the tangent function does not simply add their tangents together; rather, it combines them in a specific way that respects the cyclic nature of trigonometry.