# Math

1. Graph the following. Show all your work for full marks. You must have a scale that uses the π symbol. π¦ = βsin(π₯ βπ3)+ 2

2. Solve for 0 β€ π₯ β€ 2π. Show all your work.
π ππ2π₯ + 2πππ 2π₯ = 0

3. Use the addition/subtraction formulas to find an exact value for the following.
πππ  5π/6

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1. 1)

y = - sin ( x - Ο / 3 ) + 2

y = 2 - sin ( x - Ο / 3 )

For function:

y = A sin [ B ( x + C ) ] + D

amplitude is A

period is 2 Ο / B

phase shift is C

vertical shift is D

In this case:

A = - 1

B = 1

C = Ο / 3

D = 2

amplitude is -1

period is 2 Ο / 1 = 2 Ο

phase shift is ( Ο / 3 ) / 1 = Ο / 3

vertical shift is 2

2)

sin ( 2 x ) + 2 cos ( 2 x ) = 0

Subtract 2 cos ( 2 x ) to both sides

sin ( 2 x ) = - 2 cos ( 2 x )

Divide both sides by cos ( 2 x )

sin ( 2 x ) / cos ( 2 x ) = - 2

tan ( 2 x ) = - 2

Take the inverse tangent of both sides

2 x = Ο n + tan β» ΒΉ ( - 2 )

Since tan ( - x ) = - tan ( x )

2 x = Ο n - tan β» ΒΉ ( 2 )

Divide both sides by 2

x = 1 / 2 [ Ο n - tanβ»ΒΉ ( 2 ) ]

Since tanβ»ΒΉ ( 2 ) = 1.107148717794

x = 1 / 2 ( Ο n - 1.107148717794)

x = Ο n / 2 - 1.107148717794 / 2

x = Ο n / 2 - 0.553574359

For n = 0

x = Ο β 0 - 0.553574359 = 0 - 0.553574359 = - 0.553574359

x = - 0.553574359 radians is not in interval 0 β€ x β€ 2 Ο

For n = 1

x = Ο β 1 / 2 - 0.553574359 = Ο / 2 - 0.553574359 =

1.570796327 - 0.553574359

For n = 2

x = Ο β 2 / 2 - 0.553574359 = Ο - 0.553574359 =

3.141592654 - 0.553574359

For n = 3

x = Ο β 3 / 2 - 0.553574359 = 3 Ο / 2 - 0.553574359 =

4.712388980 - 0.553574359

For n = 4

x = Ο β 4 / 2 - 0.553574359 = 2 Ο - 0.553574359 =

6.283185307 - 0.553574359 = 5.729610948 rad

For n = 5

x = Ο β 5 / 2 - 0.553574359 = 5 Ο / 2 - 0.553574359 =

7.853981634 - 0.553574359

7.300407275 radians is not in interval 0 β€ x β€ 2 Ο

So the solutions are:

and

3)

5 Ο / 6 = 6 Ο / 6 - Ο / 6 = Ο - Ο / 6

Use identity:

cos ( Ο - ΞΈ ) = - cos ( ΞΈ )

cos ( 5 Ο / 6 ) = cos ( Ο - Ο / 6 ) = - cos ( Ο / 6 ) = - β3 / 2

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