Question

1. Graph the following. Show all your work for full marks. You must have a scale that uses the 𝜋 symbol. 𝑦 = −sin(𝑥 −𝜋3)+ 2

2. Solve for 0 ≤ 𝑥 ≤ 2𝜋. Show all your work.
𝑠𝑖𝑛2𝑥 + 2𝑐𝑜𝑠2𝑥 = 0

3. Use the addition/subtraction formulas to find an exact value for the following.
𝑐𝑜𝑠 5𝜋/6

Answer 1

1)

y = - sin ( x - π / 3 ) + 2

y = 2 - sin ( x - π / 3 )

For function:

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

amplitude is A

period is 2 π / B

phase shift is C

vertical shift is D

In this case:

A = - 1

B = 1

C = π / 3

D = 2

amplitude is -1

period is 2 π / 1 = 2 π

phase shift is ( π / 3 ) / 1 = π / 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 = π n + tan ⁻ ¹ ( - 2 )

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

2 x = π n - tan ⁻ ¹ ( 2 )

Divide both sides by 2

x = 1 / 2 [ π n - tan⁻¹ ( 2 ) ]

Since tan⁻¹ ( 2 ) = 1.107148717794

x = 1 / 2 ( π n - 1.107148717794)

x = π n / 2 - 1.107148717794 / 2

x = π n / 2 - 0.553574359


For n = 0

x = π ∙ 0 - 0.553574359 = 0 - 0.553574359 = - 0.553574359

x = - 0.553574359 radians is not in interval 0 ≤ x ≤ 2 π

For n = 1

x = π ∙ 1 / 2 - 0.553574359 = π / 2 - 0.553574359 =

1.570796327 - 0.553574359

x = 1.017221968 rad

For n = 2

x = π ∙ 2 / 2 - 0.553574359 = π - 0.553574359 =

3.141592654 - 0.553574359

x = 2.588018295 rad

For n = 3

x = π ∙ 3 / 2 - 0.553574359 = 3 π / 2 - 0.553574359 =

4.712388980 - 0.553574359

x = 4.158814621rad

For n = 4

x = π ∙ 4 / 2 - 0.553574359 = 2 π - 0.553574359 =

6.283185307 - 0.553574359 = 5.729610948 rad

For n = 5

x = π ∙ 5 / 2 - 0.553574359 = 5 π / 2 - 0.553574359 =

7.853981634 - 0.553574359

x = 7.300407275 rad

7.300407275 radians is not in interval 0 ≤ x ≤ 2 π

So the solutions are:

x = 1.017221968 rad

x = 2.588018295 rad

x = 4.158814621rad

and

x = 5.729610948 rad

3)

5 π / 6 = 6 π / 6 - π / 6 = π - π / 6

Use identity:

cos ( π - θ ) = - cos ( θ )

cos ( 5 π / 6 ) = cos ( π - π / 6 ) = - cos ( π / 6 ) = - √3 / 2