Problem 86 - Precalculus Essentials

Question

Use words to describe the formula for each of The following:the tangent of the sum of two angles.

Step-by-Step Solution

Answer

The tangent of the sum of two angles is equal to the sum of the tangent of the first angle and the tangent of the second angle, divided by 1 minus the product of the tangent of the first angle and the tangent of the second angle.

Step 1: Understanding the Tangent

The tangent of an angle in a right-angled triangle is the ratio of the side opposite the angle to the side adjacent to the angle.

Step 2: Formula for the Sum of Two Angles

The formula for the tangent of the sum of two angles, denoted as a and b, is given as the sum of the tangent of angle a and tangent of angle b divided by 1 minus the product of the tangents of angle a and angle b. This is represented as \(\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\).

Step 3: Describing the Formula in Words

To describe the above formula in words, start with 'The tangent of the sum of two angles,' which refers to \(\tan(a+b)\). Then, 'is equal to,' which translates the = symbol. Then continue with 'the sum of the tangent of the first angle and the tangent of the second angle,' referring to \(\tan(a) + \tan(b)\), followed by 'divided by 1 minus the product of the tangent of the first angle and the tangent of the second angle,' translating the denominator '1 - \tan(a)\tan(b)'.

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions.
It is often encountered in the study of right-angled triangles and is abbreviated as \(\tan\).
In a right-angled triangle, the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
This can be mathematically expressed as:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

Understanding this ratio allows us to solve problems where we need to find unknown sides or angles in triangles.
The tangent function is particularly useful because it helps relate the angle to the lengths of the sides of the triangle without needing the hypotenuse.
It's important to note that the tangent function can also be expressed using the sine and cosine functions as:

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

Angle Addition Formula

The angle addition formula provides a way to find the tangent of the sum of two angles, which is quite practical in trigonometry.
This formula is a trigonometric identity and is described as:

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

The formula helps us calculate the tangent of two angles added together without first finding individual values of the angles.
This formula can be broken down into parts:

• The numerator of the fraction shows the sum of the tangents of each angle.
• The denominator displays the subtraction of the product of the tangents from one.

By understanding this formula, you can analyze complex trigonometric problems that involve angles adding up.
It is widely applied in various fields including physics, engineering, and architecture where resolving compound angles are necessary.
Understanding this formula aids in simplifying trigonometric equations, making calculations more manageable.

Right-Angled Triangle

Right-angled triangles play a crucial role in trigonometry and are characterized by one angle of 90 degrees.
This type of triangle is the basis for defining the primary trigonometric functions: sine, cosine, and tangent.
In a right-angled triangle:

• The "hypotenuse" is the longest side and opposite the right angle.
• The "opposite" side is directly opposite the angle of interest.
• The "adjacent" side is the one next to the angle of interest.

These relationships are key to understanding how different angles and sides of a triangle interact, which is essential for solving many practical problems involving heights, distances, and angles.
With the tangent function and angle addition formula, the power of right-angled triangles can be harnessed to tackle more complex angle problems.
Recognizing the properties of right-angled triangles simplifies the process of applying trigonometric identities and resolving various calculation challenges, making it a cornerstone in both academic and real-world scenarios.