Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 154^{\circ}-\tan 49^{\circ}}{1+\tan 154^{\circ} \tan 49^{\circ}}$$

Step-by-Step Solution

Answer

\(\tan 105^{\circ}\)

Step 1: Identify the tangent of the difference of two angles formula

Recognize the given expression as the right side of the tangent of the difference of two angles formula, which is \(tan(x-y)=(tan(x)-tan(y))/(1+tan(x)tan(y))\). Here, \(x=154^{\circ}\) and \(y=49^{\circ}\).

Step 2: Implement the formula

Plug these values into the expression on the left side of the equation, which results in \(\tan(154^{\circ}-49^{\circ})\).

Step 3: Simplify the expression

Calculate the subtraction \(154^{\circ}-49^{\circ}\) which yields \(105^{\circ}\). The expression thus simplifies to \(\tan 105^{\circ}\).

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions that relates an angle in a right-angled triangle to the ratio of the length of the opposite side to the length of the adjacent side. The formula for the tangent of an angle \theta is written as \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \). In the context of the unit circle or more advanced contexts, the tangent function maps any angle to the ratio of the sine and cosine of that angle, written as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Angle Subtraction

Angle subtraction is a way to find the tangent of the difference between two angles. When an angle is described as \( x - y \), you are effectively measuring the angle obtained when angle \( y \) is subtracted from angle \( x \). Understanding angle subtraction is crucial when dealing with trigonometric identities, as it helps in simplifying expressions to a form that is much easier to evaluate or recognize. A classic application of this concept is in the tangent of the difference of two angles formula, which expresses the tangent of the difference between two angles in terms of the tangents of the individual angles.

For instance, the tangent of the difference of two angles formula is \( \tan(x - y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x)\tan(y)} \). This identity can be used to simplify complex trigonometric expressions, as it breaks down the tangent of a subtracted angle into a function of the tangents of the two separate angles.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables within their domains. They provide a way to express certain trigonometric functions in terms of others, allowing for the simplification of complex expressions and calculations. Identities are foundational to advancing in trigonometry, as they enable problem solvers to manipulate expressions that would otherwise be burdensome to solve.

One such identity is the angle difference identity for the tangent function, which is essential in exercises where you have to transform the tangent of the difference of two angles into a more workable form. By using these identities, one can resolve what looks like a difficult trigonometric problem into simpler terms. In the exercise we're exploring, the tangent difference identity is employed to convert the expression into the tangent of a single angle, thus streamlining the process of finding the solution.