Problem 9 - Precalculus with Calculus Previews

Question

Find the indicated value without the use of a calculator. $$ \tan \frac{23 \pi}{4} $$

Step-by-Step Solution

Answer

\( \tan \frac{23\pi}{4} = -1 \).

Step 1: Convert the angle to degrees

First, we need to convert the given radian angle \( \frac{23\pi}{4} \) into degrees, using the conversion factor \( 180^\circ / \pi \). \[ \text{Degrees} = \frac{23\pi}{4} \times \frac{180^\circ}{\pi} = \frac{23 \times 180^\circ}{4} = 1035^\circ \]

Step 2: Simplify the angle

Next, simplify the angle \( 1035^\circ \) by finding its equivalent in the standard position (i.e., within \( 0^\circ \) to \( 360^\circ \)). This means we take the remainder when \( 1035 \) is divided by \( 360 \).\[ 1035 \div 360 = 2 \text{ remainder } 315\]Thus, \( 1035^\circ \equiv 315^\circ \) modulo \( 360^\circ \).

Step 3: Determine the reference angle and quadrant

The angle \( 315^\circ \) lies in the fourth quadrant. In this quadrant, the tangent is negative. The reference angle is the smallest angle towards the x-axis, which is \( 360^\circ - 315^\circ = 45^\circ \). Thus, \(315^\circ \) has a reference angle of \( 45^\circ \).

Step 4: Evaluate the tangent value

Using the reference angle of \( 45^\circ \), we note that \( \tan 45^\circ = 1 \). However, since the angle is in the fourth quadrant, where the tangent is negative, \[ \tan 315^\circ = -\tan 45^\circ = -1\]

Key Concepts

Angle Conversion

When working with trigonometric functions, angles can be measured in degrees or radians. It's often necessary to convert between these two units.

In the given exercise, we start by converting the angle from radians to degrees to simplify the analysis. The conversion factor to remember is typically:

• 1 radian = 180°/π

Using this, we convert the given angle \( \frac{23\pi}{4} \) radians:

• Multiply \( \frac{23\pi}{4} \) by \( \frac{180°}{\pi} \).
• This results in \( 1035° \).

This step makes it easier to analyze the angle in a format most learners find familiar: degrees. Such conversions are fundamental in trigonometry as they allow you to work fluidly between different units of angle measurement.

Reference Angle

A reference angle is the smallest angle that a given angle makes with the x-axis. Understanding reference angles simplifies finding trigonometric values for any angle.

In this problem, after converting and simplifying the angle, we determine the reference angle for \( 315° \). To find it:

• Since \( 315° \) is in the fourth quadrant, subtract it from \( 360° \).
• The calculation goes: \( 360° - 315° = 45° \).

Thus, the reference angle for \( 315° \) is \( 45° \). This step is crucial because the trigonometric values for an angle will have the same magnitude as their reference angle, differing only by the sign based on the quadrant.

Quadrant Analysis

Quadrant analysis helps us understand the nature of trigonometric functions, such as whether they are positive or negative, based on their position on the Cartesian plane.

The plane is divided into four quadrants and for the angle calculated in our exercise \( 315° \):

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive.
• Third Quadrant: Tangent is positive.
• Fourth Quadrant: Cosine is positive.

Since \( 315° \) lies in the fourth quadrant, and knowing that cosine is positive and sine is negative there, tangent's value (sine divided by cosine) will be negative. This quadrant analysis is critical for determining the sign of the trigonometric value.

Tangent Evaluation

Now that we have our reference angle and have assessed the quadrant, we can evaluate the tangent of the angle \( 315° \). In trigonometry, it's useful to know that:

• For reference angles of \( 45° \), \( \tan 45° = 1 \).

However, since \( 315° \) is in the fourth quadrant where the tangent function is negative, the actual value becomes:\[ \tan 315° = - \tan 45° = -1 \]
This approach using reference angles and signs respective to the quadrant efficiently yields the value without needing a calculator. Understanding tangent evaluations helps in quickly determining trigonometric values across different angles.