Evaluate the following expressions. Show all work and provide the exact values (no decimals)

Question

Evaluate the following expressions. Show all work and provide the exact values (no decimals)

1. 𝒄𝒐𝒔^−𝟏 (𝒔𝒊𝒏 ( 𝟓𝝅/𝟔 ))
2. 𝒔𝒊𝒏 (𝒄𝒐𝒔^−𝟏 ( 𝟏/𝟑 ))
3. 𝒕𝒂𝒏^−𝟏 (𝒕𝒂𝒏 ( 𝟐𝝅 𝟑 ))

Answer 1

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(1.) ArcCos [ sin (5pi/6) ]

sin(5pi/6) = 1/2 Note: 5pi/6 radians = 150 degrees

ArcCos [1/2] = The angle whose cosine is 1/2

ArcCos [ 1/2 ] = pi/3 or 5*pi/3

Note: On the unit circle, there are two angles who have a cosine = 1/2

Since the cosine is positive, those angles will be in Q1 and Q4

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(2.) sin [ ArcCos [ 1/3 ]

Construct a right-triangle that has a cosine of 1/3

The adjacent side = 1 and hypotenuse = 3

So, the Cosine = 1/3

Use Pythagorean Theorem to calculate the opposite side

Opposite Side = sqrt( 3^2 - 1^1)

Opposite Side = sqrt( 8 )

Opposite Side = 2*sqrt(2)

So,

Sine = Opposite / hypotenuse

Sine = 2*sqrt(2) / 3

Therefore,

sin [ ArcCos [ 1/3 ] = 2*sqrt(2) / 3

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(3.) ArcTan [ tan(2pi/3 )]

These are inverse, or opposite, functions.

Therefore,

ArcTan [ tan(2pi/3 )] = 2pi/3