Problem 14 - Precalculus with Calculus Previews

Question

Find the indicated value without the use of a calculator. $$ \sec \frac{29 \pi}{4} $$

Step-by-Step Solution

Answer

\(-\sqrt{2}\)

Step 1: Standard Angle Conversion

To simplify the expression, convert the angle \( \frac{29 \pi}{4} \) into a standard angle by finding its equivalent angle in the interval \([0, 2\pi)\). Do this by subtracting multiples of \(2\pi\) from the angle. First, calculate \(2\pi\) in terms of \(\pi\) fractions, which is \(\frac{8\pi}{4}\). Divide 29 by 8, which gives 3 as the quotient and 5 as the remainder. Therefore, \(\frac{29\pi}{4} = \frac{(3\times8 + 5)\pi}{4} = \frac{5\pi}{4}\). So, \(\frac{29\pi}{4}\) is equivalent to \(\frac{5\pi}{4}\).

Step 2: Determine Reference Angle

Now, find the reference angle for \(\frac{5\pi}{4}\). Since \(\frac{5\pi}{4}\) is equivalent to \(\pi + \frac{\pi}{4}\), its reference angle is \(\frac{\pi}{4}\).

Step 3: Consider the Quadrant

Identify the quadrant in which \(\frac{5\pi}{4}\) lies. It lies in the third quadrant, where the secant function is negative because the cosine function is negative in this quadrant.

Step 4: Secant Value Determination

The secant function is the reciprocal of the cosine function. The cosine of the reference angle \(\frac{\pi}{4}\) is \(\frac{1}{\sqrt{2}}\). Therefore, the secant is the reciprocal: \(\sqrt{2}\). Since \(\frac{5\pi}{4}\) is in the third quadrant, the value of secant is negative, so \(\sec \frac{5\pi}{4} = -\sqrt{2}\).

Key Concepts

Standard Angle Conversion

When dealing with trigonometric functions, angles often need to be simplified into their equivalent within the standard range of [0, 2\pi). This is known as standard angle conversion. For the given angle \(\frac{29\pi}{4}\), we need to reduce it so it fits within this range. By dividing 29 by 8, we get a quotient of 3 (which corresponds to complete circles of \(2\pi\)) and a remainder of 5. The remainder, \(\frac{5\pi}{4}\), represents the equivalent angle in the standard range. Remember, the standard angle conversion simplifies calculating trigonometric functions by reducing large angles.

Reference Angle

The concept of a reference angle helps us simplify and calculate trigonometric functions for angles not on the main axis. Reference angles are always the smallest angle between the terminal side of the given angle and the horizontal axis. For \(\frac{5\pi}{4}\), this is derived as it is past \(\pi\) (or 180 degrees), standing as \(\pi + \frac{\pi}{4}\). Thus, the reference angle is \(\frac{\pi}{4}\). Using this angle helps in finding the function values, as each reference angle corresponds to known values of sine, cosine, and others, no matter the original angle's quadrant.

Quadrant Analysis

Analyzing the quadrant where an angle belongs is crucial in determining the signs of trigonometric functions. For the angle \(\frac{5\pi}{4}\), it lies in the third quadrant of the Cartesian plane. Each quadrant affects trigonometric function signs:

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine and cosecant are positive.
• Third Quadrant: Tangent and cotangent are positive.
• Fourth Quadrant: Cosine and secant are positive.

In the third quadrant, cosine is negative, so the secant, being its reciprocal, is also negative. This quadrant analysis allows us to apply the correct sign to our calculations.

Reciprocal Trigonometric Functions

Trigonometric functions have corresponding reciprocal functions that can be quite useful. The six trigonometric functions and their reciprocals include:

• Sine (\( \sin \)) and cosecant (\( \csc \)).
• Cosine (\( \cos \)) and secant (\( \sec \)).
• Tangent (\( \tan \)) and cotangent (\( \cot \)).

For our exercise with \( \sec \frac{5\pi}{4} \), the calculation involves finding the reciprocal of the cosine of the reference angle. With \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), its reciprocal is \( \sqrt{2} \). Knowing which function corresponds to which reciprocal trigonometric function can simplify recalculating angles, especially across different quadrants and angles.