Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 325^{\circ}-\tan 116^{\circ}}{1+\tan 325^{\circ} \tan 116^{\circ}}$$

Step-by-Step Solution

Answer

The given expression can be written as \( \tan(209^{\circ}) \).

Step 1: Identify the given angle values

We have two angle values, \(A = 325^{\circ}\) and \(B = 116^{\circ}\).

Step 2: Subtract angle B from angle A

To find the value of \(A - B\), perform the operation \(325^{\circ} - 116^{\circ}\). This yields a result of \(209^{\circ}\).

Step 3: Substitute the result into the tangent function

Now, substitute this result into the function \(\tan(A-B)\). This gives us \(\tan(209^{\circ})\).

Key Concepts

Angle Subtraction Formula

The angle subtraction formula is a key component of trigonometry. It helps us calculate the tangent, sine, or cosine of an angle formed by subtracting one angle from another. Specifically, for tangent, the formula is:\[ \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]This handy formula is used to simplify expressions involving tangent functions of combined angles. By breaking down a problem into simpler parts, it makes it easier to calculate the result. In the given exercise, we use it to rewrite the expression in terms of the tangent of a single angle. Knowing how to utilize such identities can transform complex trigonometric expressions into a more manageable form.

• Use the formula to combine or separate angle expressions.
• Decrease difficulty by simplifying trigonometric calculations.
• Remember, similar formulas exist for sine and cosine functions too!

By mastering the angle subtraction formula, you open a powerful toolset for solving diverse trigonometric problems. This technique not only helps in transforming expressions but also enriches your understanding of angles and their interrelationships.

Tangent Function

The tangent function is one of the primary trigonometric functions and plays a crucial role in geometry and calculus. It is defined as the ratio of the sine and cosine of an angle: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] In practical terms, it represents the slope of the line in the coordinate plane. The tangent function tends to infinity as the angle approaches an odd multiple of 90 degrees, which is a unique characteristic compared to sine and cosine.An interesting property of the tangent function, as shown in this exercise, is its application in the angle subtraction formula. Calculating \(\tan(A-B)\) reveals the angle difference's tangent by knowing individual tangents of \(A\) and \(B\). This is especially useful because direct calculation of differences in tangents via subtraction alone would not yield the correct result.

• Recognize the situations to apply tangent formulas for combining or separating angles.
• Keep in mind the periodic nature of tangents and how it affects calculations at specific intervals.

Understanding and applying the tangent function in this way makes analyzing angles far more intuitive.

Trigonometric Problem Solving

Trigonometric problem solving often involves recognizing identities and applying formulas effectively to simplify complex expressions. The exercise we examined serves as an excellent example of this approach. By using the angle subtraction formula for tangent, we were able to simplify an otherwise complicated problem involving the subtraction of two angles' tangents. The process involves a few clear steps:

• Recognize the given angles and expressions.
• Apply the appropriate trigonometric identity.
• Simplify the expression using known angles and values.

In our exercise, we first identified the angles involved, then performed an angle subtraction to find the new angle. We then applied the angle subtraction formula to express it as a simple tangent function. This showcases how familiarizing oneself with such formulas not only simplifies the process but also clarifies the path to a solution.
Trigonometric problem solving is like putting together a puzzle, where the right formulas act as the pieces that fit seamlessly, leading to the correct solution. Mastering this approach builds a strong foundation for tackling more advanced mathematical challenges.