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Chapter 6: Problem 68

Find the function value. Round to four decimal places. $$\sec 35.28^{\circ}$$

Short Answer

Expert verified
1.2249

Step by step solution

01

Understand the Secant Function

The secant function, \(\text{sec}(x)\), is the reciprocal of the cosine function, \(\text{cos}(x)\). This means that \(\text{sec}(x) = \frac{1}{\text{cos}(x)}\).
02

Calculate Cosine of the Angle

First, find the cosine of the given angle. Using a calculator in degree mode, find \( \text{cos}(35.28^{\circ}) \). This gives us approximately \( \text{cos}(35.28^{\circ}) \approx 0.8163 \).
03

Take the Reciprocal

To find the secant, take the reciprocal of the cosine value: \( \text{sec}(35.28^{\circ}) = \frac{1}{ \text{cos}(35.28^{\circ})} \). This gives us \( \text{sec}(35.28^{\circ}) \approx \frac{1}{0.8163} \approx 1.2249 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function, denoted as \( \text{sec}(x) \), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function. This means that \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \).

If you know the cosine of an angle, you can easily find the secant by taking its reciprocal. Here's a quick summary:
  • First, determine \( \text{cos}(x) \).
  • Then, take the reciprocal to get \( \text{sec}(x) \), which is \( \frac{1}{\text{cos}(x)} \).
Understanding this relationship is crucial for solving problems related to the secant function!
Cosine Function
The cosine function, \( \text{cos}(x) \), is one of the primary trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right triangle.

When you are given an angle, you can calculate its cosine using a calculator set to the degree mode. For example, for an angle of \( 35.28^{\circ} \),

\[ \text{cos}(35.28^{\circ}) \approx 0.8163 \].

This value helps us in calculating other related trigonometric functions. Knowing how to find the cosine is a fundamental step in solving many trigonometric problems.
Reciprocal
A reciprocal of a number is simply 1 divided by that number. In mathematical terms, the reciprocal of \( x \) is \( \frac{1}{x} \). This concept is widely used in trigonometry.

For example, if you have the cosine of an angle \[ \text{cos}(x) \approx 0.8163 \], to find the secant:
  • Take the reciprocal of the cosine value.
This means \[ \text{sec}(35.28^{\circ}) = \frac{1}{0.8163} \approx 1.2249 \].

Grasping the idea of reciprocals can simplify many trigonometric calculations and make solving problems much easier!

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Most popular questions from this chapter

Determine the amplitude, the period, and the phase shift of the function. Then check by graphing the function using a graphing calculator. Try to visualize the graph before creating it. $$y=5-2 \cos \left(\frac{\pi}{2} x+\frac{\pi}{2}\right)$$

Graph both functions in the same viewing window and describe how \(g\) is a transformation of $f . $$f(x)=|x|, g(x)=\frac{1}{2}|x-4|+1$$

Find all numbers \(x\) that satisfy the following. Check using a graphing calculator. a) \(\sin x=1\) b) \(\cos x=-1\) c) \(\sin x=0\)

Graph using addition of ordinates. Then check your work using a graphing calculator. $$y=\sin x+\cos 2 x$$

Convert to radian measure. Leave the answer in terms of \(\pi\). $$200^{\circ}$$
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