Problem 102 $$f(x)=\sin x, g(x)=\cos x, h(x)... [FREE SOLUTION]

91影视

Chapter 7: Problem 102

$$f(x)=\sin x, g(x)=\cos x, h(x)=\tan x, F(x)=\csc x, G(x)=\sec x, \text{ and } H(x)=\cot x$$ (a) Find $g\left(120^{\circ}\right) .$ What point is on the graph of $g ?$ (b) Find $F\left(120^{\circ}\right) .$ What point is on the graph of $F ?$ (c) Find $H\left(120^{\circ}\right)$. What point is on the graph of $H ?$

Short Answer

Expert verified

g(120鈭�) = 120鈭�

Step by step solution

Convert angle to radians

Content Microsteps text should explain exactly the same conversion method interesting

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Sine Function

The sine function, represented as $\text{sin}(x)$, is fundamental in trigonometry. It relates the angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse.

For example, if $\theta$ is an angle in a right triangle, then $\text{sin}(\theta) = \frac{opposite}{hypotenuse}$.
The function is periodic with a period of $ 2\text{蟺} $, meaning it repeats every $ 360^{\text{o}} $ or $ 2\text{蟺} \text{ radians}$.
The range of the sine function is $[-1, 1]$.

The sine of $0^{\text{o}}$ is 0, $90^{\text{o}}$ is 1, $180^{\text{o}}$ is 0, and $270^{\text{o}}$ is -1.

Cosine Function

The cosine function, or $\text{cos}(x)$, complements the sine function. It represents the ratio of the length of the adjacent side to the hypotenuse.

Hence, $\text{cos}(\theta) = \frac{adjacent}{hypotenuse}$.
Like the sine, it is periodic with a period of $2\text{蟺}$.
The range is also $[-1, 1]$.

The cosine of $0^{\text{o}}$ is 1, $90^{\text{o}}$ is 0, $180^{\text{o}}$ is -1, and $270^{\text{o}}$ is 0.

Tangent Function

The tangent function, written as $\text{tan}(x)$, combines both sine and cosine. It is the ratio of $\text{sin}(x)$ to $\text{cos}(x)$.

Mathematically, $ \text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} $.
The function is periodic with a period of $\text{蟺}$.
Its range is $ (-\text{鈭瀩, \text{鈭瀩)$.

Basic values include $\text{tan}(0^{\text{o}}) = 0$, $\text{tan}(45^{\text{o}}) = 1$, and $\text{tan}(90^{\text{o}})$ is undefined due to division by zero.

Cosecant Function

The cosecant function, denoted as $\text{csc}(x)$, is the reciprocal of the sine function. This means, $\text{csc}(x) = \frac{1}{\text{sin}(x)}$.

It is undefined wherever $\text{sin}(x)=0$.
The function's period is $2\text{蟺}$.
Its range is $(-\text{鈭瀩, -1] \cup [1, \text{鈭瀩)$.

As an example, $\text{csc}(30^{\text{o}}) = 2$ because $\text{sin}(30^{\text{o}})= \frac{1}{2}$.

Secant Function

Similar to the cosecant function, the secant function $\text{sec}(x)$ is the reciprocal of the cosine function. Thus, $\text{sec}(x) = \frac{1}{\text{cos}(x)}$.

It is undefined wherever $\text{cos}(x)=0$.
The secant function has a period of $2\text{蟺}$.
Its range is $(-\text{鈭瀩, -1] \cup [1, \text{鈭瀩)$.

For instance, $\text{sec}(60^{\text{o}}) = 2$ because $\text{cos}(60^{\text{o}}) = \frac{1}{2}$.

Cotangent Function

The cotangent function $\text{cot}(x)$ is the reciprocal of the tangent function: $ \text{cot}(x) = \frac{1}{\text{tan}(x)} $.

It is much like dividing $\text{cos}(x)$ by $\text{sin}(x)$.
The function is undefined wherever $\text{tan}(x) = 0$.
Its period is $\text{蟺}$, similar to the tangent function.
The range can take any real value $( -\text{鈭瀩, \text{鈭瀩 )$.

For instance, $\text{cot}(45^{\text{o}}) = 1$ as $\text{tan}(45^{\text{o}}) = 1$.

Degree to Radian Conversion

To convert between degrees and radians, the following relations are used:

$1^{\text{o}} = \frac{\text{蟺}}{180} \text{ radians}$.
Conversely, $1 \text{ radian} = \frac{180}{\text{蟺}}^{\text{o}}$.

For the exercise:
The conversion of $120^{\text{o}}$ to radians is

$120^{\text{掳}} \times \frac{\text{蟺}}{180^{\text{掳}}} = \frac{120\text{蟺}}{180} = \frac{2\text{蟺}}{3} \text{ radians}$.

Understanding these conversions is essential for working with trigonometric functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Convert angle to radians

Key Concepts

Sine Function

Cosine Function

Tangent Function

Cosecant Function

Secant Function

Cotangent Function

Degree to Radian Conversion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Geometry

Mechanics Maths

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.