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Discuss why this statement is incorrect: arccos ( cos x ) = x for all x .

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Determine whether the following statement is true or false and explain your answer: arccos ( x ) = π arccos x .

True . The angle, θ 1 that equals arccos ( x ) , x > 0 , will be a second quadrant angle with reference angle, θ 2 , where θ 2 equals arccos x , x > 0 . Since θ 2 is the reference angle for θ 1 , θ 2 = π θ 1 and arccos ( x ) = π arccos x -

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Algebraic

For the following exercises, evaluate the expressions.

sin 1 ( 1 2 )

π 6

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cos 1 ( 2 2 )

3 π 4

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tan 1 ( 3 )

π 3

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For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

cos 1 ( 0.4 )

1.98

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For the following exercises, find the angle θ in the given right triangle. Round answers to the nearest hundredth.

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

sin 1 ( cos ( π ) )

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tan 1 ( sin ( π ) )

0

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cos 1 ( sin ( π 3 ) )

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tan 1 ( sin ( π 3 ) )

0.71

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sin 1 ( cos ( π 2 ) )

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tan 1 ( sin ( 4 π 3 ) )

-0.71

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sin 1 ( sin ( 5 π 6 ) )

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tan 1 ( sin ( 5 π 2 ) )

π 4

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cos ( sin 1 ( 4 5 ) )

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sin ( cos 1 ( 3 5 ) )

0.8

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sin ( tan 1 ( 4 3 ) )

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cos ( tan 1 ( 12 5 ) )

5 13

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cos ( sin 1 ( 1 2 ) )

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For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle.

tan ( sin 1 ( x 1 ) )

x 1 x 2 + 2 x

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sin ( cos 1 ( 1 x ) )

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cos ( sin 1 ( 1 x ) )

x 2 1 x

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cos ( tan 1 ( 3 x 1 ) )

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tan ( sin 1 ( x + 1 2 ) )

x + 0.5 x 2 x + 3 4

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Extensions

For the following exercises, evaluate the expression without using a calculator. Give the exact value.

sin 1 ( 1 2 ) cos 1 ( 2 2 ) + sin 1 ( 3 2 ) cos 1 ( 1 ) cos 1 ( 3 2 ) sin 1 ( 2 2 ) + cos 1 ( 1 2 ) sin 1 ( 0 )

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For the following exercises, find the function if sin t = x x + 1 .

cos ( sin 1 ( x x + 1 ) )

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tan 1 ( x 2 x + 1 )

t

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Graphical

Graph y = sin 1 x and state the domain and range of the function.

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Graph y = arccos x and state the domain and range of the function.

A graph of the function arc cosine of x over -1 to 1. The range of the function is 0 to pi.

domain [ 1 , 1 ] ; range [ 0 , π ]

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Graph one cycle of y = tan 1 x and state the domain and range of the function.

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For what value of x does sin x = sin 1 x ? Use a graphing calculator to approximate the answer.

approximately x = 0.00

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For what value of x does cos x = cos 1 x ? Use a graphing calculator to approximate the answer.

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Real-world applications

Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-floor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?

0.395 radians

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Suppose you drive 0.6 miles on a road so that the vertical distance changes from 0 to 150 feet. What is the angle of elevation of the road?

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An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

1.11 radians

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Without using a calculator, approximate the value of arctan ( 10 , 000 ) . Explain why your answer is reasonable.

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A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

1.25 radians

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Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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