How do you find the exact circular function value of csc(-23pi/6)? I don't know how to do the pi symbol on the keyboard.
first find the smallest positive coterminal angle to your given angle
-23π/6 = -24π/6 + π/6 --- >
so csc(-23π/6) = csc π/6
= 1/sin π/6
= 1/(1/2)
= 2
check with your calculator, (make sure you are set to radians)
To find the exact circular function value of csc(-23π/6), let's break down the steps:
1. Converting -23π/6 to degrees:
- Remember that π radians is equal to 180 degrees.
- To convert from radians to degrees, multiply the value by (180/π).
So, -23π/6 degrees = -23π/6 * (180/π) degrees = -23 * (180/6) degrees = -23 * 30 degrees = -690 degrees.
2. Finding the reference angle:
- Since the given angle is negative, we need to add 360 degrees to determine its equivalent positive angle.
- Reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.
-690 degrees + 360 degrees = -330 degrees.
The reference angle is 330 degrees.
3. Determining the quadrant:
- The reference angle falls in the fourth quadrant, as it is greater than 180 degrees and less than 270 degrees.
4. Evaluating the circular function:
- csc(θ) is the reciprocal of sin(θ).
- In the fourth quadrant, sin(θ) is positive.
- Therefore, sin(330 degrees) = sin(-330 degrees) = sin(30 degrees).
5. Using the unit circle or trigonometric identities:
- The unit circle shows that sin(30 degrees) = 1/2.
6. Finding the reciprocal:
- Since csc(θ) is the reciprocal of sin(θ), csc(30 degrees) = 1/(1/2) = 2.
Therefore, the exact circular function value of csc(-23π/6) is 2.
To find the exact circular function value of csc(-23π/6), we need to understand the concept of circular functions and their values.
The circular functions (sine, cosine, tangent, cosecant, secant, and cotangent) are commonly used in mathematics to study the relationships and properties of angles in a circle. These functions can be found on the unit circle.
Now, let's find the exact value of csc(-23π/6):
1. Start by converting -23π/6 into degrees. Since π radians is equal to 180 degrees, we can multiply -23π/6 by (180/π) to convert it into degrees.
-23π/6 * (180/π) = -23 * 30 = -690 degrees
2. Next, we need to find the reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis in the standard position.
To find the reference angle, we can remove or add full rotations (360 degrees) until the angle is within 0 to 360 degrees or 0 to 2π radians. In this case, we need to add 360 degrees to -690 degrees until we get a positive value within 0 to 360 degrees.
-690 + 360 = -330 degrees
So, the reference angle is 330 degrees.
3. Now, we can find the value of csc(330 degrees) on the unit circle. The cosecant (csc) function is the reciprocal of the sine function.
On the unit circle, the sine function is positive in the second and third quadrants. The value of the sine function at 330 degrees is the same as the value at its reference angle, which is 30 degrees.
On the unit circle, at 30 degrees, the sine value is 1/2. Therefore, the csc value will be the reciprocal of 1/2, which is 2.
So, the exact circular function value of csc(-23π/6) is 2.
Please note that if you don't know how to type the π symbol on the keyboard, you can use "pi" or "Pi" in its place. Just make sure to indicate in your work that you are referring to π equivalent to the ratio of the circumference to the diameter of a circle.