Problem 98 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\tan \frac{4 \pi}{3}$$

Step-by-Step Solution

Answer

\( \tan \frac{4\pi}{3} = \sqrt{3} \) with decimal approximation 1.732.

Step 1: Determine Reference Angle

To find the reference angle for the angle \( \frac{4\pi}{3} \), we note that it is in the third quadrant (since \( \pi < \frac{4\pi}{3} < \frac{3\pi}{2} \)). The reference angle \( \theta\) can be calculated as:\[ \theta = \frac{4\pi}{3} - \pi = \frac{\pi}{3} \] So, the reference angle is \( \frac{\pi}{3} \).

Step 2: Express in Terms of Reference Angle

The tangent function in the third quadrant is positive. Thus, the tangent of \( \frac{4\pi}{3} \) can be expressed using its reference angle \( \frac{\pi}{3} \):\[ \tan \frac{4\pi}{3} = \tan \frac{\pi}{3} \]

Step 3: Find Exact Value of Tangent of Reference Angle

The exact value of \( \tan \frac{\pi}{3} \) is known from trigonometric identities:\[ \tan \frac{\pi}{3} = \sqrt{3} \]

Step 4: Verify with Calculator

Now, use a calculator to find the decimal approximation.Calculate \( \tan \frac{4\pi}{3} \) using a calculator:Decimal approximation: \( \tan \frac{4\pi}{3} \approx 1.732 \)The exact value \( \sqrt{3} \) also approximates to 1.732, confirming that both the exact value and calculator approximation of \( \tan \frac{4\pi}{3} \) are the same.

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions, often abbreviated as "tan." It relates the angles in a right triangle to the ratio of the opposite side to the adjacent side. In a circle, the tangent function can be defined using the coordinates of points on the unit circle. The tangent of an angle \( \theta \) is given by the ratio of the \( y \)-coordinate to the \( x \)-coordinate of the point on the unit circle corresponding to that angle.

Key properties of the tangent function include:

• Periodicity: The tangent function is periodic with a period of \( \pi \), meaning that \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \).
• Symmetry: It is an odd function, so \( \tan(-\theta) = -\tan(\theta) \). This means it is symmetric about the origin.
• Undefined Points: The tangent function is undefined for angles \( \frac{\pi}{2} + k\pi \), where \( k \) is an integer, because at these angles, the \( x \)-coordinate is zero, resulting in division by zero.

Understanding these properties helps in graphing the tangent function and solving trigonometric equations.

Reference Angle

A reference angle is a useful concept in trigonometry that helps simplify the calculation of trigonometric functions. A reference angle is the smallest angle that a given angle makes with the x-axis. It is always measured as a positive acute angle, no greater than \( \frac{\pi}{2} \) (or 90 degrees).

To find a reference angle for any given angle, follow these steps:

• For angles in the first quadrant: The reference angle is the angle itself.
• For angles in the second quadrant: Subtract the angle from \( \pi \) (or 180 degrees).
• For angles in the third quadrant: Subtract \( \pi \) from the angle.
• For angles in the fourth quadrant: Subtract the angle from \( 2\pi \) (or 360 degrees).

Using a reference angle allows you to determine the values of trigonometric functions for angles in any quadrant by evaluating the function of the reference angle, then applying appropriate signs based on the quadrant.

Exact Value Calculation

Exact value calculations involve determining the specific, non-approximate values of trigonometric functions for certain key angles. These angles are typically ones that result in mathematically simple results, which are memorized from trigonometric identities. Common angles include multiples of \( \pi/6, \pi/4, \text{and} \pi/3 \).

Understanding exact values is crucial in trigonometry because they:

• Help simplify expressions and avoid calculator-based round-off errors.
• Provide a clear understanding of the relationship between different trigonometric functions.
• Are used in solving trigonometric equations analytically.

For example, for the reference angle \( \frac{\pi}{3} \), the exact value of the tangent function is \( \tan(\frac{\pi}{3}) = \sqrt{3} \). Using these kinds of exact values can greatly support the problem-solving process in trigonometry.

Third Quadrant Angles

Third quadrant angles are angles that, when placed on the standard position on the unit circle, fall into the third quadrant. This quadrant lies between \( \pi \) and \( \frac{3\pi}{2} \) radians (or 180 to 270 degrees). For these angles, both the sine and cosine values are negative, which results in a positive tangent value since tangent is the ratio of sine to cosine.

When working with third quadrant angles, it is essential to remember:

• Sign of Functions: \( \sin \theta \) and \( \cos \theta \) are both negative, thus \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is positive.
• Finding Reference Angles: To find the reference angle, subtract \( \pi \) from the given angle if in radians.

Recognizing these characteristics helps to simplify problems involving trigonometric functions in different quadrants. This simplification is crucial in both analytic and graphical representations of trigonometric functions.