Problem 95 - 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). $$\tan \frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

1. \(-\tan(\frac{\pi}{4})\), 2. -1, 3. Calculator gives -1.

Step 1: Identify the Reference Angle

First, determine the reference angle for \( \frac{3\pi}{4} \). Since \( \frac{3\pi}{4} \) is in the second quadrant, the reference angle is found by subtracting \( \frac{3\pi}{4} \) from \( \pi \). Thus, the reference angle is \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).

Step 2: Write the Function in Terms of the Reference Angle

In the second quadrant, the tangent function is negative. Therefore, \( \tan\left(\frac{3\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) \).

Step 3: Calculate the Exact Value

The exact value of \( \tan\left(\frac{\pi}{4}\right) \) is 1. Therefore, \( \tan\left(\frac{3\pi}{4}\right) = -1 \).

Step 4: Use a Calculator to Verify the Decimal Approximation

Using a calculator, find the decimal approximation of \( \tan\left(\frac{3\pi}{4}\right) \). It should be approximately -1. Similarly, the approximate value for \( -1 \) is exactly -1. This confirms the result.

Key Concepts

Reference Angle

In trigonometry, a reference angle is the acute angle that a given angle makes with the x-axis. It is always between 0 and \( \frac{\pi}{2} \) or 0 and 90 degrees, even if the original angle isn’t. To find the reference angle for any given angle, especially those measured in radians, you consider the angle’s position in the coordinate system and calculate accordingly.

• For angles in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from \( \pi \).
• For the third quadrant, subtract \( \pi \) from the angle.
• In the fourth quadrant, subtract the angle from \( 2\pi \).

For example, for the angle \( \frac{3\pi}{4} \), since it is in the second quadrant, we subtract it from \( \pi \) to get the reference angle: \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \). This simple calculation helps in simplifying the original trigonometric function.

Tangent Function

The tangent function, denoted as \( \tan \), is a key trigonometric function related to angles in a right triangle. It gives the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Tangent also has periodic properties with a period of \( \pi \). This means the same values repeat for every \( \pi \) radians or 180 degrees. It can take any real number as its value. The function is positive in the first and third quadrants and negative in the second and fourth quadrants.
To evaluate \( \tan(\frac{3\pi}{4}) \) using the reference angle \( \frac{\pi}{4} \), we note that in the second quadrant, tangent values are negative. Thus, \( \tan\left(\frac{3\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1 \). This showcases how understanding the behavior of tangent in different quadrants can simplify calculations.

Trigonometric Quadrants

The concept of trigonometric quadrants is foundational in understanding how trigonometric functions behave. The entire Cartesian plane is divided into four quadrants, each characterized by certain trigonometric properties. These quadrants are defined by the angle at which they lie relative to the x-axis.

• First Quadrant (0 to \( \frac{\pi}{2} \)): All trigonometric functions are positive.
• Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \)): Sine is positive, cosine and tangent are negative.
• Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \)): Tangent is positive, sine and cosine are negative.
• Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \)): Cosine is positive, sine and tangent are negative.

Contour these rules with the position of \( \frac{3\pi}{4} \) in the second quadrant where the tangent function yields a negative value. It emphasizes the importance of knowing your angle's quadrant to accurately determine the sign of your trigonometric outcomes.