/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Evaluate the sine, cosine, and t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 53

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$225^{\circ}$$

Short Answer

Expert verified
\(\sin(225掳) = -\frac{1}{\sqrt{2}}\), \(\cos(225掳) = -\frac{1}{\sqrt{2}}\), and \(\tan(225掳) = 1\).

Step by step solution

01

Identify Quadrant

In this case, the angle 225掳 lies in quadrant III in the coordinate plane (since the 180掳 < angle < 270掳). In this quadrant, sine and cosine are negative, while tangent is positive. Then, we need to identify equivalent commonly used angles, so express 225掳 as 180掳 + 45掳.
02

Calculate Sine of the Angle

We'll start with the sine function. From the standard values table, we know that \( \sin(45掳) \) is \( \frac{1}{\sqrt{2}} \). Given that in quadrant III, sine is negative, we get: \( \sin(225掳) = -\sin(45掳) = -\frac{1}{\sqrt{2}} \).
03

Calculate Cosine of the Angle

Now, we'll go to the cosine function. From the standard values table, we know that \( \cos(45掳) \) is \( \frac{1}{\sqrt{2}} \). Since cosine is also negative in the third quadrant we get: \( \cos(225掳) = -\cos(45掳) = -\frac{1}{\sqrt{2}} \).
04

Calculate Tangent of the Angle

Finally, we'll find the tangent. The tangent function is given by \( \tan(胃) = \frac{\sin(胃)}{\cos(胃)} \). Substituting , we get: \( \tan(225掳) = \frac{\sin(225掳)}{\cos(225掳)} = \frac{-\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}} = 1 \), as tangent is positive in quadrant III.

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, denoted as \(\sin(胃)\), is a fundamental trigonometric function that relates the angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. One way to remember the sine function is by the acronym SOH, which stands for "Sine = Opposite over Hypotenuse." For reference angles like \(45掳\), the sine value is commonly known to be \( \frac{1}{\sqrt{2}} \).

When dealing with angles beyond \(180掳\), such as \(225掳\), it is important to consider the quadrant in which the angle lies. For \(225掳\), this is the third quadrant where the values of sine are negative. Hence, \(\sin(225掳) = -\frac{1}{\sqrt{2}}\).

Understanding which quadrant an angle falls into helps in determining the sign and value of the sine function effectively.
Cosine Function
The cosine function is another critical trigonometric function, symbolized by \(\cos(胃)\). It measures the ratio of the adjacent side to the hypotenuse of a right-angled triangle. A handy memory aid is the acronym CAH, which means "Cosine = Adjacent over Hypotenuse." The cosine of \(45掳\) is \( \frac{1}{\sqrt{2}} \).

Like the sine function, the cosine value also depends on the quadrant the angle is in. For \(225掳\), which lies in the third quadrant, cosine values are negative. Consequently, \(\cos(225掳) = -\frac{1}{\sqrt{2}}\).

Quadrants play a vital role in determining the positivity or negativity of the cosine value for angles beyond \(90掳\) degrees. Remembering which operations to perform based on the angle's quadrant helps in quickly evaluating cosine.
Tangent Function
Tangent, represented as \(\tan(胃)\), is integral to trigonometry. It corresponds to the ratio of the sine to the cosine of an angle. Thus, \(\tan(胃) = \frac{\sin(胃)}{\cos(胃)}\). In a basic right triangle, tangent expresses the ratio of the opposite side to the adjacent side (TOA - "Tangent = Opposite over Adjacent").

For an angle like \(225掳\), in the third quadrant, while sine and cosine are both negative, their ratio (tangent) is positive. This results in \(\tan(225掳) = 1\) given that dividing two identical negative numbers yields a positive outcome.

The sign of the tangent function is influenced directly by the quadrant because it relies on both sine and cosine. Quadrant awareness is crucial for correctly determining multiplication outcomes and signs for tangent values.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

The Leaning Tower of Pisa is not vertical, but when you know the angle of elevation \(\theta\) to the top of the tower as you stand \(d\) feet away from it, you can find its height \(h\) using the formula \(h=d \tan \theta\).

The normal monthly high temperatures \(H\) (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $$H(t)=56.94-20.86 \cos \left(\frac{\pi t}{6}\right)-11.58 \sin \left(\frac{\pi t}{6}\right)$$ and the normal monthly low temperatures \(L\) are approximated by $$L(t)=41.80-17.13 \cos \left(\frac{\pi t}{6}\right)-13.39 \sin \left(\frac{\pi t}{6}\right)$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January (see figure). (Source: National Climatic Data Center) (a) What is the period of each function? (b) During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sky around June \(21,\) but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

Fill in the blank. If not possible, state the reason. As \(x \rightarrow-1^{+},\) the value of arcsin \(x \rightarrow\) \(\square\).

Determine whether the statement is true or false. Justify your answer. $$\arctan x=\frac{\arcsin x}{\arccos x}$$

The daily consumption \(C\) (in gallons) of diesel fuel on a farm is modeled by $$C=30.3+21.6 \sin \left(\frac{2 \pi t}{365}+10.9\right)$$ where \(t\) is the time (in days), with \(t=1\) corresponding to January 1. (a) What is the period of the model? Is it what you expected? Explain. (b) What is the average daily fuel consumption? Which term of the model did you use? Explain. (c) Use a graphing utility to graph the model. Use the graph to approximate the time of the year when consumption exceeds 40 gallons per day.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.