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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

(1+cosA)(1-cosA)=sin^2A
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(1+cosA)(1-cosA)=sin^2A
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Catherine
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kurash
Solve Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
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questions
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AjA
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
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Kuz
A = P(1 + r/n) ^rt
Dale
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What is the value of log-1
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Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
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Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
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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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