Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Express \(\tan (u+v+w)\) in terms of trigonometric functions of \(u, v\), and \(w\).

Step-by-Step Solution

Answer

\(\tan(u+v+w) = \frac{\tan u + \tan v + \tan w - \tan u \tan v \tan w}{1 - \tan u \tan v - \tan v \tan w - \tan u \tan w}\)

Step 1: Use the Identity for Two Angles

First, recall the identity for \( \tan(a + b) \): \[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]We will utilize this identity to express \(\tan(u+v+w)\) in terms of \(\tan u\), \(\tan v\), and \(\tan w\) in stages.

Step 2: Apply Identity to \(\tan(u+v)\)

Applying the identity to \(\tan(u+v)\), we get:\[\tan(u+v) = \frac{\tan u + \tan v}{1 - \tan u \tan v}\]This expression gives us the tangent of \(u+v\) in terms of \(\tan u\) and \(\tan v\).

Step 3: Express \(\tan(u+v+w)\) Using \(\tan(u+v)\) and \(\tan w\)

Now, substitute \(\tan(u+v)\) and \(\tan w\) into the two-angle tangent identity:\[\tan(u+v+w) = \frac{\tan(u+v) + \tan w}{1 - \tan(u+v) \tan w}\]We already have \(\tan(u+v)\) from Step 2, so substitute that in:\[\tan(u+v+w) = \frac{\frac{\tan u + \tan v}{1 - \tan u \tan v} + \tan w}{1 - \left(\frac{\tan u + \tan v}{1 - \tan u \tan v}\right) \tan w}\]

Step 4: Simplify the Expression

To simplify the expression for \(\tan(u+v+w)\), compute the numerator and the denominator separately:**Numerator**:\[(\tan u + \tan v + \tan w(1 - \tan u \tan v))\]**Denominator**:\[((1 - \tan u \tan v) - (\tan u + \tan v) \tan w)\]The expression for \(\tan(u+v+w)\) becomes:\[\tan(u+v+w) = \frac{\tan u + \tan v + \tan w - \tan u \tan v \tan w}{1 - \tan u \tan v - \tan v \tan w - \tan u \tan w}\]

Step 5: Conclusion

We arrived at the expression for \(\tan(u+v+w)\) using the formula for the tangent of a sum of angles:\[\tan(u+v+w) = \frac{\tan u + \tan v + \tan w - \tan u \tan v \tan w}{1 - \tan u \tan v - \tan v \tan w - \tan u \tan w}\]

Key Concepts

Sum of Angles Formula

Trigonometric identities can be incredibly useful when we're dealing with complex angles.
One important concept is the "sum of angles formula." This formula helps us find the tangent or sine of a sum of two or three angles using the tangents or sines of individual angles.
As a reminder, the formula for the sum of two angles in terms of tangent is:

• \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \)

When you need to add three angles, like in our exercise, you build up to it step-by-step.
You first calculate the sum of two angles, then add the third one using the same concept. This method of gradually building with these identities helps break down complicated problems into simpler parts, making them easier to understand and solve.

Tangent Function

The tangent function is a crucial part of trigonometry and, along with sine and cosine, is a fundamental trigonometric function.
The tangent of an angle, often abbreviated as \( \tan \), represents the ratio between the sine and cosine of an angle:

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

This means that the tangent gives a measure of the slope or steepness of a line connecting the origin to a point on the unit circle.
In practical terms, tangent can describe angles in various scenarios, from simple triangle problems to complex wave functions.
For solving sums of angles, knowing that tangent relates to sine and cosine helps us utilize identities like the sum of angles formula effectively, as it allows us to connect different trigonometric properties seamlessly.

Angle Addition Identity

The angle addition identity is a powerful tool that allows us to combine angles in trigonometric expressions efficiently.
When dealing with tangent, the angle addition identity formula becomes instrumental:

• \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \)

This identity helps us because it shows how to find the tangent of the sum of two angles using individual tangents.
It's especially useful in our exercise for simplifying expressions like \( \tan(u+v+w) \).
By applying this identity incrementally, first to \( u+v \) and then to add \( w \), we build up the solution progressively.
The angle addition identity is essential because it allows us to manage complex angles by breaking them into simpler, more manageable parts, helping us solve equations and understand geometric and wave relationships.