/*! 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 131 Find the exact value of each fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 131

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-270^{\circ}$$

Short Answer

Expert verified
(a) 1, (b) -1, (c) 0, (d) 0, (e) 0, (f) 0.

Step by step solution

01

Calculate (f+g)(胃)

Substitute 胃 with -270掳 in f(胃)=sin胃 and g(胃)=cos胃. \n f(-270掳) = sin(-270掳) = 1 (since the sine of -270掳 is 1), and g(-270掳) = cos(-270掳) = 0 (since the cosine of -270掳 is 0). Then calculate (f+g)(-270掳) which is 1 + 0 = 1.
02

Calculate (g-f)(胃)

We use the values calculated in the previous step. Then, the difference (g - f) (-270掳) becomes 0 - 1 = -1.
03

Calculate [g(胃)]^2

Square g(-270掳) which is 0. So, [g(-270掳)]^2 = 0^2 = 0.
04

Calculate (fg)(胃)

Multiply the results of f(胃) and g(胃). So, (fg)(-270掳) becomes 1 * 0 = 0.
05

Calculate f(2胃)

Substitute 2胃 with 2(-270掳)=-540掳 in f(胃). Remembering the period of sine is 360掳, we simplify 鈭�540掳 to 鈭�540掳 + 360掳*2 = 180掳. Therefore, f(-540掳)=sin(180掳)=0.
06

Calculate g(-胃)

Substitute 鈭捨� with -(-270掳)=270掳 into g(胃). Remember that cosine is an even function which means cos(胃) = cos(鈭捨�). Hence, g(-(-270掳))=cos(270掳)=0.

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 by \( \sin \theta \), is a fundamental trigonometric function that measures the y-coordinate of a point on the unit circle as it relates to a given angle. The sine function repeats every 360 degrees, which means it has a period of 360掳.

This cyclic nature of sine can be observed in its wave-like graph, oscillating between -1 and 1. It's important to remember:
  • At 0掳 and 180掳, \( \sin \theta \) is 0.
  • It reaches its maximum value of 1 at 90掳 and minimum of -1 at 270掳.
  • The sine of negative angles can be calculated using \( \sin(-\theta) = -\sin(\theta) \).
If you are trying to calculate \( \sin(-270掳) \), keep in mind these principles, especially that the sine of -270掳 equals 1.
Cosine Function
The cosine function, denoted as \( \cos \theta \), is another key trigonometric function. It corresponds to the x-coordinate of a point on the unit circle. The cycle of cosine, just like sine, repeats every 360 degrees. Cosine is distinguished by:
  • Being 1 at 0掳, and -1 at 180掳.
  • Having zeros at 90掳 and 270掳.
  • Being an even function, \( \cos(\theta) = \cos(-\theta) \).
In the problem, for \( \cos(-270掳) \), it equates to 0. This is due to the symmetrical nature of the cosine function along the y-axis.
Exact Values
Exact values refer to specific outcomes of trigonometric functions without using a calculator, typically at standard angles such as 0掳, 30掳, 45掳, 60掳, 90掳, and their multiples. These can be derived from the unit circle and fundamental identities such as Pythagorean identities.

Knowing these values can help compute expressions swiftly and is often tackled by memorizing or understanding triangles and their angles. For example:
  • \( \sin(90掳) = 1 \), \( \cos(90掳) = 0 \).
  • \( \sin(180掳) = 0 \), \( \cos(180掳) = -1 \).
  • Symmetry and periodicity help in deriving values for negative angles or ones beyond 360掳.
In problems like the one given, compute exact values such as \( \sin(-270掳) \) and \( \cos(-270掳) \) by leveraging these known values and patterns.
Angle Conversion
Angle conversion is essential for dealing with trigonometric functions, particularly when angles are not in standard form. The process involves changing angles into measures easier to evaluate, such as converting a negative angle or one larger than 360掳.

This can be done by:
  • Adding or subtracting multiples of 360掳 to bring angles within the 0掳 to 360掳 range.
  • Using known angle identities to simplify calculations.
  • Knowing that, for sine and cosine, the repeating pattern helps to compute angles larger than 360掳 efficiently.
For example, converting \( -540掳 \) into an equivalent angle, you would add 720掳 (2*360掳), resulting in \( 180掳 \). Understanding this helps when calculating functions like \( \sin(-540掳) \) by simplifying them into recognizable angles.

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

True or False Determine whether the statement is true or false. Justify your answer. Writing Find two bearings perpendicular to \(\mathrm{N} 32^{\circ} \mathrm{E}\) and explain how you found them.

A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is 16 feet from the streetlight and 5 feet from the tip of the streetlight's shadow, the person's shadow starts to appear beyond the streetlight's shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities and use a variable to indicate the height of the streetlight. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the streetlight?

Sketch the graph of \(y=\cos b x\) for \(b=\frac{1}{2}, 2\) and \(3 .\) How does the value of \(b\) affect the graph? How many complete cycles of the graph of \(y\) occur between 0 and \(2 \pi\) for each value of \(b ?\)

(a) Use a graphing utility to complete the table. Round your results to four decimal places. $$\begin{array}{|l|l|l|l|l|l|}\hline \theta & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} \\\\\hline \sin \theta & & & & & \\\\\hline \cos \theta & & & & & \\\\\hline \tan \theta & & & & & \\\\\hline\end{array}$$ (b) Classify each of the three trigonometric functions as increasing or decreasing for the table values. (c) From the values in the table, verify that the tangent function is the quotient of the sine and cosine functions.

Determine whether the statement is true or false. Justify your answer. $$\sin 60^{\circ} \csc 60^{\circ}=1$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete 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.