Solve for all possible values of x where O°≤x360°.
sin x=2/3.
I do not understand these types of questions please answer in as much detail as you can so I can do them myself.
The following reference solves a similar question in a way that applies to the current question.
http://www.jiskha.com/display.cgi?id=1310912979
Note that for the current question, there are two quadrants that satisfy the sign requirement, namely first and second quadrants.
To find θ you will need to add or subtract the reference angle t from the horizontal line, namely 0° or 180°.
If you need further help, please post.
To find all possible values of x where O° ≤ x ≤ 360° and sin(x) = 2/3, you can follow the steps below:
Step 1: Determine the principal angle
Since the sin(x) value is positive, we know that the angle x is in either the first or second quadrant. The principal angle is the angle whose sin value is equal to 2/3. To find this angle, you can use the inverse sin function (sin^(-1)) on a calculator or reference table.
sin^(-1)(2/3) ≈ 41.81°
So, the principal angle is approximately 41.81°.
Step 2: Determine the reference angle
In the first quadrant, the reference angle is equal to the principal angle. However, in the second quadrant, the reference angle is equal to 180° minus the principal angle.
For the given sin(x) = 2/3, the reference angle in the second quadrant is:
180° - 41.81° ≈ 138.19°
Step 3: Find the solutions in the first and second quadrants
In the first quadrant, the angle x is the same as the principal angle:
x = 41.81°
In the second quadrant, the angle x can be found by adding the reference angle to 180°:
x = 180° + 138.19° ≈ 318.19°
So, the possible values of x in the range 0° ≤ x ≤ 360° where sin(x) = 2/3 are approximately 41.81° and 318.19°.
Note: There are infinitely many solutions for sin(x) = 2/3 since trigonometric functions are periodic. However, we confined the solutions to the range of 0° to 360° as specified in the question.
To solve the equation sin x = 2/3, we need to find all possible values of x that satisfy this equation within the given range O° ≤ x ≤ 360°.
Step 1: Understand the equation.
The equation sin x = 2/3 tells us that the sine of angle x is equal to 2/3. The sine function represents the ratio of the length of the opposite side of a right triangle to the length of its hypotenuse. In this case, we are looking for angles whose sine ratio is 2/3.
Step 2: Determine the reference angle.
To find the reference angle, we can use the inverse sine function (sin⁻¹), also known as arcsine. The inverse sine function takes the ratio of the opposite side to the hypotenuse and returns the angle.
sin⁻¹(2/3) = 41.81°
Step 3: Find the solutions in the given range.
Since the sine function is periodic, i.e., it repeats every 360°, we can find all the solutions within the given range by adding or subtracting multiples of the reference angle.
Since the given range is O° ≤ x ≤ 360°, we can use the reference angle as a base and find the solutions by adding and subtracting multiples of 360°:
x = 41.81° ± n(360°), where n is an integer.
Using this formula, we can calculate the values of x that satisfy the equation within the given range. Here are a few examples:
For n = 0:
x = 41.81° + 0(360°) = 41.81°
For n = 1:
x = 41.81° + 1(360°) = 401.81°
For n = -1:
x = 41.81° - 1(360°) = -318.19° (note that -318.19° is equivalent to 41.81° counterclockwise from 360°)
You can continue this process for any value of n that satisfies O° ≤ x ≤ 360° to find all possible values of x that satisfy the equation.