Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate without using a calculator. (a) \(\tan \frac{\pi}{6}\) (b) \(\sec \pi\) (c) \(\sec \frac{3 \pi}{4}\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{3}}{3}\); (b) \(-1\); (c) \(-\sqrt{2}\).

Step 1: Understanding Trigonometric Ratios

To solve the given exercise, we'll evaluate each trigonometric function individually, starting with understanding the basic trigonometric ratios and their respective angles on the unit circle.

Step 1: Evaluate \( \tan \frac{\pi}{6} \)

The angle \( \frac{\pi}{6} \) radians is equivalent to 30 degrees. The tangent of 30 degrees is a well-known trigonometric ratio, \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \). Simplifying gives \( \frac{\sqrt{3}}{3} \).

Step 2: Evaluate \( \sec \pi \)

The angle \( \pi \) radians is 180 degrees. The secant function is the reciprocal of the cosine function, \( \sec \theta = \frac{1}{\cos \theta} \). Since \( \cos \pi = -1 \), it follows that \( \sec \pi = \frac{1}{-1} = -1 \).

Step 3: Evaluate \( \sec \frac{3\pi}{4} \)

The angle \( \frac{3\pi}{4} \) radians corresponds to 135 degrees. In the unit circle, the cosine of \( \frac{3\pi}{4} \) is \( -\frac{\sqrt{2}}{2} \). Thus, the secant is the reciprocal, \( \sec \frac{3\pi}{4} = -\frac{1}{\frac{\sqrt{2}}{2}} = -\sqrt{2} \).

Key Concepts

Unit Circle

When studying trigonometric functions, the unit circle is a vital tool. It's a circle with a radius of one, centered at the origin of a coordinate plane. This circle helps us define trigonometric functions for all angles. By using the unit circle, we can easily find the sine, cosine, and tangent of any angle by observing where that angle intercepts the circle. For example, each angle on the unit circle corresponds to a specific point \( (x, y) \). Here \( x \) represents the cosine value, and \( y \) represents the sine value of the angle.
Furthermore, the unit circle helps in visualizing angles beyond 90 degrees by extending angles counter-clockwise and including full 360-degree rotations. This visual and mathematical tool provides a deep understanding of how trigonometric functions behave and interact with angles.

Radians to Degrees Conversion

Radians and degrees are two units used to measure angles. Radians often offer convenience when dealing with natural mathematical operations and calculus. At the same time, degrees can be more intuitive for understanding everyday angles. To convert radians to degrees, we use the conversion factor where \( \pi \) radians equal 180 degrees. This gives us a simple formula:

• Degrees = Radians \( \times \) \( \frac{180}{\pi} \).
• Radians = Degrees \( \times \) \( \frac{\pi}{180} \).

For example, in the exercise provided, \( \frac{\pi}{6} \) radians converts to 30 degrees. Using this formula facilitates understanding and calculating trigonometric functions in different units, whether utilizing radians or degrees.

Tangent Function

The tangent of an angle is one of the primary trigonometric functions. It is the ratio of the sine of that angle to its cosine. In the context of the unit circle and a right triangle, it can be expressed as:

• \( \tan(\theta) = \frac{\text{sine}(\theta)}{\text{cosine}(\theta)} \).

Geometrically, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side.
For example, for \( \tan(\frac{\pi}{6}) \), you first convert the angle to degrees (30 degrees) and find that its tangent value is \( \frac{\sqrt{3}}{3} \). It shows how \( \tan \) can reveal different fundamental properties of an angle's trigonometric context. Unlike sine and cosine, tangent can have values greater than one, and it also can take any real value.

Secant Function

Secant is another important trigonometric function, closely related to the cosine. It is the reciprocal of the cosine function, defined as:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

This function gives us the ratio of the hypotenuse to the adjacent side in a right triangle.
In the exercise, we evaluate \( \sec(\pi) \) and know that \( \cos(\pi) = -1 \), so the secant, being the reciprocal, is \(-1\). Similarly, \( \sec(\frac{3\pi}{4}) \) corresponds to 135 degrees, where we find \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \). Therefore, its secant value will be \(-\sqrt{2}\).
Understanding secant is crucial since it can also have values greater than one and undefined when cosine equals zero, due to division.