Problem 65 - Algebra and Trigonometry

Question

use reference angles to find the exact value of each expression. Do not use a calculator. $$ \tan 420^{\circ} $$

Step-by-Step Solution

Answer

\(\tan(420^\circ) = \sqrt{3}\)

Step 1: Find the equivalent angle in the first rotation

Start by subtracting 360 degrees from 420 degrees as many times as necessary until reach an angle that falls within the first full rotation. For this instance, if we subtract 360 degrees from 420 degrees once, we get 60 degrees. Thus, the angle equivalent to 420 degrees within the first rotation is 60 degrees.

Step 2: Determine the exact value of the tangent

With the equivalent angle within a single rotation (0 ≤ angle < 360 degree) known, use it to find the exact value of the tangent function without a calculator. In this case, we have the angle 60 degrees. Recall that \(\tan(60^\circ) = \sqrt{3}\). Thus, \(\tan(420^\circ) = \tan(60^\circ) = \sqrt{3}\).

Step 3: Check the results

Finally, check if the results make sense compared with your prior knowledge about the tangent function. In this particular context, where tangent is being evaluated for the angle of 420 degrees, it is expected to get an exact value since 420 degrees corresponds to an angle in the first quadrant, where the tangent function is positive. Also, by the periodic property of the tangent function, tan(420) equals tan(60), which equals sqrt(3).

Key Concepts

Understanding Reference Angles

To solve trigonometric problems like finding the exact value of \( \tan(420^\circ) \), the concept of reference angles becomes extremely handy. A reference angle is the smallest angle measurement between the terminal arm of a given angle and the x-axis.

Here are some key points about reference angles:

• Reference angles are always measured in positive degrees.
• The reference angle for any angle within one rotation (0° to 360°) is always between 0° and 90°.
• To find the reference angle for an angle greater than 360°, keep subtracting 360° until you fall into the first rotation (0° - 360°).

In our exercise, 420° was beyond one full rotation. Therefore, we subtracted 360° and found the equivalent angle of 60°, which lies in the first quadrant. The reference angle is particularly useful because it allows us to apply known trigonometric values of common angles like 30°, 45°, and 60°, to angles on the unit circle.

Exploring the Tangent Function

The tangent function is one of the primary trigonometric functions, often denoted as \( \tan \). It relates the ratios of the sides of a right-angled triangle. Specifically, in a triangle with angle \( \theta \), tangent is the ratio of the opposite side to the adjacent side.

Some critical aspects of the tangent function are:

• The formula is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• The tangent of 60° is a special value, \( \tan(60^\circ) = \sqrt{3} \).
• Tangent is positive in the first and third quadrants, and negative in the second and fourth quadrants.

In our exercise, once we reduced the angle 420° to an equivalent 60° angle, finding the tangent became straightforward. Since 60° is in the first quadrant, where tangent values are positive, we concluded that \( \tan(420^\circ) = \tan(60^\circ) = \sqrt{3} \).

Periodic Properties of Functions

Trigonometric functions, such as the tangent function, are periodic. This means they repeat their values in regular intervals, allowing for consistency and predictability across multiple rotations on the unit circle.

Key points about these periodic properties include:

• The period of the tangent function is 180° or \( \pi \) radians.
• This means \( \tan(\theta + 180^\circ) = \tan(\theta) \).
• Due to this periodicity, the function produces the same value every half rotation, which makes it particularly useful for calculations involving angles beyond 360°.

In the specific case of our exercise, the periodic nature of tangent ensured that \( \tan(420^\circ) \) was equivalent to \( \tan(60^\circ) \), as 420° is equivalent to 60° plus a full rotation of 360°. This periodic property lets us predict function values without recalculating from scratch.