Problem 17 - Algebra and Trigonometry

Question

\(13-18\) m Use an addition or subtraction formula to write the expression as a trigonometric function of one number, and then find its exact value. $$ \frac{\tan 73^{\circ}-\tan 13^{\circ}}{1+\tan 73^{\circ} \tan 13^{\circ}} $$

Step-by-Step Solution

Answer

The exact value is \( \sqrt{3} \).

Step 1: Identify the Formula

The expression can be simplified using the tangent subtraction formula: \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \). In this case, \( a = 73^{\circ} \) and \( b = 13^{\circ} \).

Step 2: Apply the Formula

Using the identified formula, we rewrite the given expression as \( \tan(73^{\circ} - 13^{\circ}) \).

Step 3: Simplify the Angle Expression

Calculate \( 73^{\circ} - 13^{\circ} = 60^{\circ} \). Therefore, the expression simplifies to \( \tan(60^{\circ}) \).

Step 4: Evaluate the Trigonometric Function

Find the exact value of \( \tan(60^{\circ}) \). We know \( \tan(60^{\circ}) = \sqrt{3} \).

Key Concepts

Tangent Subtraction Formula

The tangent subtraction formula is a useful tool in trigonometry for simplifying expressions that involve the difference of angles. The formula is given by: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]This equation allows you to express the tangent of the difference between two angles, \(a\) and \(b\), as a function of their individual tangents.

• The numerator, \(\tan a - \tan b\), accounts for the difference in the tangents of the angles.
• The denominator, \(1 + \tan a \tan b\), ensures the formula considers the interaction between the angles when combined.

In our exercise, we're given angles \(a = 73^{\circ}\) and \(b = 13^{\circ}\). By substituting these values into the tangent subtraction formula, we transform the expression into a single computational step involving a simpler angle.

Exact Trigonometric Values

Exact trigonometric values play a vital role in simplifying trigonometric expressions. These are well-known values of trigonometric functions for specific angles that make solving expressions straightforward. For instance, common angles like \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\) have trigonometric values that are frequently used.
In the given problem, once we apply the tangent subtraction formula, we arrive at \(\tan(60^{\circ})\). The exact value of \(\tan(60^{\circ})\) is \(\sqrt{3}\). This value is derived from the properties of an equilateral triangle, where dividing it creates a \(30^{\circ}, 60^{\circ}\) right triangle. Recognizing these exact values is crucial for solving problems quickly and efficiently, without resorting to calculators or complex computations.

Angle Subtraction in Trigonometry

Angle subtraction in trigonometry is an essential concept when evaluating expressions that involve the difference between two angles. Understanding how angles combine or differ requires knowledge of specific trigonometric identities, such as the tangent subtraction formula.
When we subtract angles, like \(73^{\circ} - 13^{\circ}\), we're often aiming to simplify a more complex problem. Subtraction can reveal a common or special angle, such as \(60^{\circ}\), where trigonometric functions have simple, known values.

• Subtraction helps to reduce problems to manageable pieces by breaking them into simpler, more recognizable parts.
• It allows for the application of exact trigonometric values, which leads to accurate results without approximation.

Grasping angle subtraction's role can deepen our understanding of trigonometry, making it a valuable skill for solving trigonometric equations and identities.