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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

answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
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Nokwanda
C'est comment
Abdel
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SORIE
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Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
Practice Key Terms 6

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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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