How to find the exact value of cosecant of pi/3?

Introduction:

Trigonometry deals with the relationships between the angles and sides of triangles. It helps us understand various phenomena involving angles and distances. The trigonometric functions, such as sine, cosine, and tangent, play a crucial role in these calculations. In this article, we will focus on finding the exact value of the cosecant of π/3 and explore related frequently asked questions.

Understanding Cosecant:

Cosecant (csc) is the reciprocal of the sine function. It is defined as:

cscθ = 1 / sinθ

Therefore, to find the exact value of the cosecant of π/3, we need to first determine the sine of π/3.

The Answer: How to Find the Exact Value of Cosecant of π/3?

To find the exact value of the cosecant of π/3, let’s start by finding the sine of π/3:

Step 1: Recall the unit circle. π/3 corresponds to an angle of 60 degrees, which lies in the first quadrant.
Step 2: In the unit circle, draw a radius forming an angle of 60 degrees with the positive x-axis.
Step 3: The x-coordinate of the point where the radius intersects the unit circle gives us the sine value. In this case, the x-coordinate is 1/2.
Step 4: Hence, sin(π/3) = 1/2.

Now that we know sin(π/3) = 1/2, we can easily find the exact value of the cosecant of π/3:

csc(π/3) = 1 / sin(π/3) = 1 / (1/2) = 2.

**Therefore, the exact value of the cosecant of π/3 is 2.**

Frequently Asked Questions:

1. How do you simplify csc π/3?

To simplify csc π/3, you need to know that cscθ is the reciprocal of sinθ. In this case, sin π/3 = 1/2, so csc π/3 = 1 / (1/2) = 2.

2. What is the cosecant of 60 degrees?

Since 60 degrees is equivalent to π/3, the cosecant of 60 degrees is also 2.

3. How can I find the cosecant of any angle?

To find the cosecant of any angle, you first need to determine the sine of that angle and then take the reciprocal of the sine value.

4. What is the cosecant function?

The cosecant function, denoted as csc, is the reciprocal of the sine function. It calculates the ratio between the hypotenuse and the opposite side in a right-angled triangle.

5. What is the reciprocal of 1/2?

The reciprocal of 1/2 is 2.

6. Can the cosecant be negative?

Yes, the cosecant can be negative. It is negative in quadrants II and III in the unit circle, where the y-coordinate is negative.

7. Is the cosecant always greater than or equal to 1?

No, the cosecant can take any positive or negative value. It is equal to 1 when the sine is equal to 1, but in other cases, it can be greater or lower.

8. How do I calculate the sin of π/3?

By drawing the unit circle and considering the x-coordinate of the point where the radius intersects the unit circle, you can determine that sin(π/3) = 1/2.

9. Is the sin of π/3 equal to the cosecant of π/3?

No, the sin of π/3 is equal to 1/2, while the cosecant of π/3 is 2. They are reciprocals of each other.

10. Can the cosecant of an angle be undefined?

Yes, the cosecant of an angle is undefined when the sine of that angle is zero. For example, the cosecant of 0 degrees and 180 degrees is undefined since sin(0) = sin(180) = 0.

11. How is the unit circle helpful in finding trigonometric values?

The unit circle provides a visual representation of angles and their corresponding trigonometric values. It simplifies calculations and helps determine the exact values of the trigonometric functions.

12. What are some other important trigonometric identities?

Some other important trigonometric identities include cosine (cos), tangent (tan), secant (sec), cotangent (cot), and their respective reciprocals. These functions are extensively used in various mathematical and scientific contexts.

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