Question

Select one or more expressions that together represent all solutions to the equation 2sin(4x)+6=5.

a. −150^∘n⋅360^∘
b. −37.5^∘ n⋅90^∘
c. −30^∘ n⋅180^∘
d. −7.5^∘ n⋅90^∘
e. 7.5^∘ n⋅180^∘
f. 150^∘ n⋅360^∘

Answer 1

All solutions to the equation 2sin(4x)+6=5 can be represented by the third and fourth quadrant angles at multiples of 360 degrees or 180 degrees. The general solutions are -37.5° n· 90° and -7.5° n· 90° after dividing the common angle by 4.

Step-by-step explanation:

The student's question is asking for all solutions to the trigonometric equation 2sin(4x)+6=5. To find the solutions, we first isolate sin(4x):

1. 2sin(4x) + 6 = 5
2. 2sin(4x) = 5 - 6
3. 2sin(4x) = -1
4. sin(4x) = -1/2

We know that the sine function has a value of -1/2 at specific reference angles. Considering the periodic nature of the sine function, solutions occur at angles where 4x corresponds to an angle with sine -1/2, which happens in the third and fourth quadrants of the unit circle at multiples of 360 degrees or 180 degrees.

So, the general solutions for 4x can be written as:

• 4x = 180°n + (-150°)
• 4x = 360°n - 150°

Dividing by 4 to solve for x gives:

1. x = 45°n + (-37.5°)
2. x = 90°n - 37.5°

Therefore, the expressions that represent all solutions are:

• b. -37.5° n· 90°
• d. -7.5° n· 90°