Fine the complete exact value of sin x = -sqrt3/2.
I'm very lost as to how to even begin this problem.
Do I have to solve the right side of the equation first or bring it over to the left side?
If you're looking for a value for x, then you want to find all angles x where
sin x = -√3/2
Now, you should know that sin 60° = √3/2
sin x is negative in QII and QIII, so all solutions 0 <= x < 360 are x=180±60°. That would be 120° and 240°.
Now, sin x is periodic with period 360°, so you can add any multiple of 360° to those angles and the value of sin x is still -√3/2.
So, the complete solution is
x = 180±60° + 360n°, where n is any integer.
Oops. My bad. sin x is negative in QIII and QIV, so
x = 240° or 300°
add any multiple of 360° to those values.
To find the complete exact value of sin(x) = -√3/2, we can start by looking at the unit circle. The unit circle is a circle with a radius of 1, centered at the origin (0,0) in a coordinate plane. It is used in Trigonometry to connect the angles and trigonometric functions.
In this case, we need to find an angle, let's call it θ, such that sin(θ) = -√3/2. This means we are looking for an angle whose sine value is -√3/2, and this value corresponds to a y-coordinate on the unit circle.
The key step is to understand the special angles on the unit circle. These special angles have exact values for their trigonometric functions. For example, at 30 degrees or π/6 radians, sin(30°) = 1/2, which is a special value.
Since we have a negative value for the sine, we know that the angle should be in the third or fourth quadrant, where the y-coordinate (sine value) is negative.
To find the exact value, we need to identify an angle where the y-coordinate is -√3/2. In this case, the angle we are looking for is 240° or 4π/3 radians.
So, sin(240°) = -√3/2.
To verify this, you can use a scientific calculator, trigonometric tables, or software to find the sine of 240°. Alternatively, you can use the properties of the unit circle to find the exact sine value.
I hope this explanation helps you understand how to find the complete exact value of sin(x) = -√3/2.