Problem 36 - Precalculus Mathematics for Calculus

Question

Find the reference number for each value of \(t.\) (a) \(t=\frac{5 \pi}{6}\) (b) \(t=\frac{7 \pi}{6}\) (c) \(t=\frac{11 \pi}{3}\) (d) \(t=-\frac{7 \pi}{4}\)

Step-by-Step Solution

Answer

Reference angles: (a) \( \frac{\pi}{6} \), (b) \( \frac{\pi}{6} \), (c) \( \frac{\pi}{3} \), (d) \( \frac{\pi}{4} \).

Step 1: Understanding Reference Angles

The reference angle of any angle is the smallest angle it makes with the x-axis. It is always between 0 and \( \frac{\pi}{2} \) (or equivalent in degrees), hence always positive.

Step 2: Find Reference Angle for Part (a)

For \( t = \frac{5 \pi}{6} \), this angle is in the second quadrant. The reference angle \( \theta \) is found by calculating \(\pi - \frac{5 \pi}{6} = \frac{\pi}{6} \). Thus, the reference angle is \( \frac{\pi}{6} \).

Step 3: Find Reference Angle for Part (b)

For \( t = \frac{7 \pi}{6} \), this angle is in the third quadrant. The reference angle \( \theta \) is \( \frac{7 \pi}{6} - \pi = \frac{\pi}{6} \). Thus, the reference angle is \( \frac{\pi}{6} \).

Step 4: Normalize for Part (c)

First normalize \( t = \frac{11 \pi}{3} \) by subtracting \( 2\pi \) until it is within the range \( 0 \) to \( 2\pi \). This gives \(\frac{11 \pi}{3} - 2\pi = \frac{11 \pi}{3} - \frac{6 \pi}{3} = \frac{5 \pi}{3} \).

Step 5: Find Reference Angle for Part (c)

\( \frac{5 \pi}{3} \) is in the fourth quadrant. The reference angle is \( 2\pi - \frac{5 \pi}{3} = \frac{\pi}{3} \). Thus, the reference angle is \( \frac{\pi}{3} \).

Step 6: Normalize for Part (d)

First normalize \( t = -\frac{7 \pi}{4} \) by adding \( 2\pi \) until it is between \( 0 \) and \( 2\pi \). Adding \( 2\pi = \frac{8\pi}{4} \) gives \( -\frac{7 \pi}{4} + \frac{8 \pi}{4} = \frac{\pi}{4} \).

Step 7: Find Reference Angle for Part (d)

Since \( \frac{\pi}{4} \) is already a positive angle in the first quadrant, the reference angle is \( \frac{\pi}{4} \).

Key Concepts

Trigonometric Quadrants

The coordinate plane is divided into four distinct regions called quadrants, each serving as a home for specific angle ranges. These quadrants help us categorize angles based on their position relative to the x-axis and y-axis.
Understanding the placement of an angle among the quadrants is essential for determining the sign of trigonometric functions like sine and cosine.

• First Quadrant: Angles here measure between 0 and \( \frac{\pi}{2} \). Both sine and cosine are positive.
• Second Quadrant: Covers angles from \( \frac{\pi}{2} \) to \( \pi \), where sine is positive and cosine is negative.
• Third Quadrant: Angles range from \( \pi \) to \( \frac{3\pi}{2} \). Here, both sine and cosine are negative.
• Fourth Quadrant: Encompasses angles from \( \frac{3\pi}{2} \) to \( 2\pi \). Sine is negative, but cosine is positive.

Knowing which quadrant an angle is in can easily guide you in calculating reference angles and evaluating trigonometric functions effectively.

Angle Normalization

Angle normalization helps in simplifying angles so they fit within a standard range, typically between 0 and \( 2\pi \) radians (or 0 to 360 degrees). This is crucial in trigonometry since many functions have behavior that repeats in cycles.
Here's how you do it:

• For a given positive angle, repeatedly subtract \( 2\pi \) until it falls within the desired range.
• For negative angles, add \( 2\pi \) until the angle is positive and fits within 0 to \( 2\pi \).

This method ensures the angle is "normalized" and aligns with the standard circle, making it easier to find trigonometric function values or a reference angle.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians are based on the circle's circumference.One full circle is \( 2\pi \) radians, and a half circle (or straight angle) is simply \( \pi \) radians. This measurement system is preferred in mathematics because it relates more naturally to various formulas and equations involving circles, making calculations more intuitive.
Converting between degrees and radians is straightforward:

• To convert degrees to radians, multiply by \( \frac{\pi}{180} \) .
• To convert radians to degrees, multiply by \( \frac{180}{\pi} \) .

Understanding radians can significantly enhance your ability to solve problems involving trigonometric functions and circular motion.

Trigonometric Functions

Trigonometric functions, like sine, cosine, and tangent, form the foundation of trigonometry. They relate the angles of a triangle to the lengths of its sides but are also fundamental in describing periodic phenomena such as sound waves, light waves, and more. Here's a brief overview:

• Sine (sin): Represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (cos): Indicates the ratio of the adjacent side to the hypotenuse.
• Tangent (tan): The ratio of the opposite side to the adjacent side, essentially sine divided by cosine.

These functions have unique characteristics based on the quadrants where the angles are located, affecting their values. They are cyclical, repeating every \( 2\pi \) radians, which is why angle normalization becomes critical.These functions expand far beyond triangles, linking to concepts in calculus, physics, and engineering.