Problem 124 - Calculus Volume 1

Question

Evaluate the following functional values. $$\tan \left(\frac{19 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The value of \( \tan \left(\frac{19\pi}{4}\right) \) is \(-1\).

Step 1: Understand the Periodicity of Tangent

The tangent function, \( \tan(x) \), is periodic with a period of \( \pi \). This means that \( \tan(x) = \tan(x + \pi k) \) where \( k \) is any integer. This property allows us to find equivalent angles within the range \( [0, \pi) \).

Step 2: Simplify the Angle

To simplify \( \tan \left(\frac{19\pi}{4}\right) \), we first need to express the angle in a simpler form using the periodicity of tangent. We do this by subtracting integer multiples of \( \pi \) from \( \frac{19\pi}{4} \) until we are left with an angle within the standard period. \[\frac{19\pi}{4} = 4\pi + \frac{3\pi}{4}\]Subtract \( 4\pi \) (which is \( 4 \times \pi \)) from \( \frac{19\pi}{4} \), resulting in \( \frac{3\pi}{4} \).

Step 3: Evaluate the Equivalent Angle

Now that we've reduced the angle to \( \frac{3\pi}{4} \), we can evaluate \( \tan \left(\frac{3\pi}{4}\right) \). Recall that \( \frac{3\pi}{4} \) is in the second quadrant where tangent is negative. The reference angle here is \( \pi/4 \), which has a tangent of 1. So, \( \tan \left(\frac{3\pi}{4}\right) = -1 \).

Key Concepts

Periodicity of Tangent

The periodicity of a function speaks to how often its values repeat. For the tangent function, this repetition occurs every \( \pi \). This is a unique feature compared to sine and cosine functions, which have a period of \( 2\pi \). Essentially, this means if you know \( \tan(x) \), you also inherently know \( \tan(x + k\pi) \) for any integer \( k \). Every \( \pi \) radians, the tangent function starts its cycle anew.
In practical terms, this principle allows us to reduce larger angles, like \( \frac{19\pi}{4} \), to a simpler equivalent within the first \( \pi \) radians by simply subtracting multiples of \( \pi \) until we reach the desired range.

• Start by identifying the angle \( x \).
• Determine multiples of \( \pi \) to subtract from \( x \).
• Simplify \( x \) to be within the range of \( [0, \pi) \).

This property is particularly useful, allowing for easier calculations and applicable problem solving in trigonometry.

Reference Angle

A reference angle is your go-to for breaking down complex angle problems. It's always the smallest angle to the x-axis, typically found in the first quadrant, aiding in simplifications.
For its calculation, first determine the absolute difference of your angle from the x-axis, which can simplify finding various trigonometric values. Regardless of which quadrant your actual angle is in, the sine, cosine, and tangent values at your reference angle give a direct clue about these values in the actual position by noting signs.

• In the first quadrant, angles are positive for sine, cosine, and tangent.
• In the second quadrant, sine remains positive while cosine and tangent become negative.
• In the third quadrant, tangent switches back to positive, but sine and cosine are both negative.
• In the fourth quadrant, cosine is positive, with sine and tangent being negative.

For the angle in question, \( \frac{19\pi}{4} \), reduction shows its reference angle is \( \frac{\pi}{4} \). This knowledge helps determine the tangent, which, as \( -1 \) in the second quadrant, corresponds to \( \tan(\pi/4) \) being 1.

Second Quadrant

The second quadrant in the unit circle spans angles from \( \frac{\pi}{2} \) to \( \pi \) radians. In this area, only the sine function retains its positivity. Both cosine and tangent are negative in this domain.
Understanding how angles transfer to secondary quadrants involves memorizing how each trigonometric function's sign alternates from the first quadrant. This inherently simplifies problems by allowing you to focus on determining the reference angle, knowing that all values in this domain behave predictably.

• Sine remains positive.
• Cosine turns negative.
• Tangent also becomes negative.

For instance, when solving for \( \tan \left(\frac{3\pi}{4}\right)\), it's crucial to know that the reference angle \( \frac{\pi}{4} \) typically gives a tangent result of 1. However, due to the negative value characteristic in the second quadrant, \( \tan \left(\frac{3\pi}{4}\right)\) results in \( -1 \). This quick recall of quadrant properties aids in efficiently solving trigonometry problems.