Math

1. Graph the following. Show all your work for full marks. You must have a scale that uses the πœ‹ symbol. 𝑦 = βˆ’sin(π‘₯ βˆ’πœ‹3)+ 2

2. Solve for 0 ≀ π‘₯ ≀ 2πœ‹. Show all your work.
𝑠𝑖𝑛2π‘₯ + 2π‘π‘œπ‘ 2π‘₯ = 0

3. Use the addition/subtraction formulas to find an exact value for the following.
π‘π‘œπ‘  5πœ‹/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 ⁻ ΒΉ ( - 2 )

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

    2 x = Ο€ n - tan ⁻ ΒΉ ( 2 )

    Divide both sides by 2

    x = 1 / 2 [ Ο€ n - tan⁻¹ ( 2 ) ]

    Since tan⁻¹ ( 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€ βˆ™ 0 - 0.553574359 = 0 - 0.553574359 = - 0.553574359

    x = - 0.553574359 radians is not in interval 0 ≀ x ≀ 2 Ο€

    For n = 1

    x = Ο€ βˆ™ 1 / 2 - 0.553574359 = Ο€ / 2 - 0.553574359 =

    1.570796327 - 0.553574359

    x = 1.017221968 rad

    For n = 2

    x = Ο€ βˆ™ 2 / 2 - 0.553574359 = Ο€ - 0.553574359 =

    3.141592654 - 0.553574359

    x = 2.588018295 rad

    For n = 3

    x = Ο€ βˆ™ 3 / 2 - 0.553574359 = 3 Ο€ / 2 - 0.553574359 =

    4.712388980 - 0.553574359

    x = 4.158814621rad

    For n = 4

    x = Ο€ βˆ™ 4 / 2 - 0.553574359 = 2 Ο€ - 0.553574359 =

    6.283185307 - 0.553574359 = 5.729610948 rad

    For n = 5

    x = Ο€ βˆ™ 5 / 2 - 0.553574359 = 5 Ο€ / 2 - 0.553574359 =

    7.853981634 - 0.553574359

    x = 7.300407275 rad

    7.300407275 radians is not in interval 0 ≀ x ≀ 2 Ο€

    So the solutions are:

    x = 1.017221968 rad

    x = 2.588018295 rad

    x = 4.158814621rad

    and

    x = 5.729610948 rad

    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 ) = - √3 / 2

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