Problem 32 - Precalculus: Functions and Graphs

Question

Approximate to four decimal places, when appropriate. (a) \(\sin (-0.11)\) (b) \(\sec \frac{31}{27}\) (c) \(\cos (-8.54)\) (d) \(\tan (3 \pi / 7)\)

Step-by-Step Solution

Answer

(a) -0.1094, (b) 1.1797, (c) -0.7807, (d) -0.4816

Step 1: Approximating \( \sin(-0.11) \)

Since sine is an odd function, \( \sin(-x) = -\sin(x) \). Therefore, \( \sin(-0.11) = -\sin(0.11) \). Use a calculator to find \( \sin(0.11) \), which is approximately \( 0.1094 \). Thus, \( \sin(-0.11) \) is approximately \(-0.1094\).

Step 2: Approximating \( \sec \frac{31}{27} \)

The secant function is the reciprocal of the cosine function, so \( \sec(x) = \frac{1}{\cos(x)} \). First, calculate \( \cos(\frac{31}{27}) \). Using a calculator, \( \cos(\frac{31}{27}) \approx 0.8480 \). Therefore, \( \sec(\frac{31}{27}) \approx \frac{1}{0.8480} \approx 1.1797 \).

Step 3: Approximating \( \cos(-8.54) \)

Cosine is an even function, meaning that \( \cos(-x) = \cos(x) \). Therefore, \( \cos(-8.54) = \cos(8.54) \). Calculate \( \cos(8.54) \) using a calculator, which is approximately \( -0.7807 \).

Step 4: Approximating \( \tan \left( \frac{3\pi}{7} \right) \)

Use a calculator to find \( \tan \left( \frac{3\pi}{7} \right) \). First, convert \( \frac{3\pi}{7} \) into radians if necessary and compute. The result is approximately \( -0.4816 \).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that relates an angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. In more advanced mathematics, it extends to functions in the unit circle. Sine is known to be an "odd function," which means its graph is symmetric about the origin. This property indicates that \( \sin(-x) = -\sin(x) \).

For example, when solving \( \sin(-0.11) \), you can apply the odd function rule: \( \sin(-0.11) = -\sin(0.11) \). This allows you to compute the sine of a positive angle instead, which can be easier with calculators, giving \( \sin(0.11) \approx 0.1094 \). Hence, \( \sin(-0.11) \approx -0.1094 \).

• Key property: Sine is an odd function.
• Periodic with period \(2\pi\).
• Useful in sine wave analysis and oscillatory motions.

Cosine Function

The cosine function, closely related to sine, measures the ratio of the adjacent side of a right triangle to the hypotenuse in a right triangle. In the context of the unit circle, cosine gives the x-coordinate for an angle. Unlike sine, cosine is an "even function," meaning that its graph is symmetric about the y-axis. This leads to the property \( \cos(-x) = \cos(x) \).

This even property is particularly useful in computations, such as determining \( \cos(-8.54) \). You can simply compute \( \cos(8.54) \) without changing the sign, which approximately equals \(-0.7807\).

• Key property: Cosine is an even function.
• Periodic with period \(2\pi\).
• Critical in defining the phase of waveforms.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include secant, cosecant, and cotangent. Each is the reciprocal of one of the basic trigonometric functions. For example, the secant function \( \sec(x) \) is the reciprocal of the cosine function: \( \sec(x) = \frac{1}{\cos(x)} \).

This reciprocal relation helps when calculating values that aren't easily computed directly. To find \( \sec(\frac{31}{27}) \), you first determine \( \cos(\frac{31}{27}) \), which is approximately \( 0.8480 \). Then, \( \sec(\frac{31}{27}) \approx \frac{1}{0.8480} \approx 1.1797 \).

• Includes \( \sec(x) = \frac{1}{\cos(x)} \), \( \csc(x) = \frac{1}{\sin(x)} \), and \( \cot(x) = \frac{1}{\tan(x)} \).
• Used when solving equations involving divisions of sides in triangles.
• Important for inverse trigonometric identities.

Tangent Function

The tangent function is among the most well-known trigonometric functions. It's defined as the ratio of the sine function to the cosine function: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

The tangent function is periodic, with a period of \( \pi \), unlike sine and cosine which have a period of \(2\pi\). This function does not have symmetry like sine or cosine but instead exhibits a repeating pattern. When solving for tangent values, such as \( \tan(\frac{3\pi}{7}) \), you can directly use a calculator to find it approximately \( -0.4816 \).

• Periodic with period \( \pi \).
• Helps in slope calculations and angle measurements in coordinate systems.
• Non-symmetric, with asymptotes where \( \cos(x) = 0 \).