How to find the exact value of cosecant of 3pi/4?

Cosecant is one of the fundamental trigonometric functions that can be used to express the relationship between the sides and angles of a right triangle. It is defined as the reciprocal of the sine function. Finding the exact value of cosecant of a given angle can be challenging, especially when the angle is not a special angle. In this article, we will explore how to find the exact value of cosecant of 3π/4 and provide solutions to related FAQs.

How to Find the Exact Value of Cosecant of 3π/4?

To find the exact value of cosecant (csc) of 3π/4, we need to understand the trigonometric ratios of the angle 3π/4.

The angle 3π/4 is equivalent to 135 degrees or a reference angle of π/4 in the second quadrant.

Considering a right triangle with an angle of 135 degrees, we can determine the values of the sides of the triangle:

In the second quadrant, the x-coordinate of the corresponding point on the unit circle is negative, while the y-coordinate is positive. Using the Pythagorean theorem, we can find the values of the sides:

Cosine (cos) of 3π/4: Since the adjacent side is -1 and the hypotenuse is √2, we can calculate cos(3π/4) = -1/√2 = -√2/2.

Sine (sin) of 3π/4: Since the opposite side is 1 and the hypotenuse is √2, we can calculate sin(3π/4) = 1/√2 = √2/2.

Reciprocal of the sine function yields the cosecant (csc) function:

Cosecant (csc) of 3π/4 = 1 / sin(3π/4) = 1 / (√2/2) = 2 / √2 = √2.

Thus, the exact value of cosecant of 3π/4 is √2.

Now that we have determined the exact value of cosecant of 3π/4, let’s address some related FAQs:

FAQs:

1. What is 3π/4 in degrees?

Angle 3π/4 is equivalent to 135 degrees.

2. How can I find the reference angle of 3π/4?

The reference angle of 3π/4 is π/4.

3. What does the cosecant function represent?

The cosecant function represents the ratio between the hypotenuse and the opposite side of a right triangle.

4. How can I find the exact value of cosecant of a given angle?

To find the exact value of cosecant of a given angle, you need to determine the values of the sides of a right triangle corresponding to that angle and then calculate the reciprocal of the sine function.

5. What are the trigonometric ratios in the second quadrant?

In the second quadrant, the sine function is positive, while the cosine and tangent functions are negative.

6. Can the cosecant function be negative?

Yes, the cosecant function can be negative in certain quadrants.

7. How does the value of cos(θ) relate to the value of sin(θ)?

The value of cos(θ) is equal to sin(π/2 – θ) for any angle θ.

8. What is the relationship between the sine and cosecant functions?

The relationship between the sine and cosecant functions is that the cosecant function is the reciprocal of the sine function.

9. How can I find the sine function of 3π/4?

To find the sine function of 3π/4, you can use the values of the sides of a right triangle corresponding to that angle and calculate the ratio of the opposite side to the hypotenuse.

10. What is the reciprocal of a fraction?

The reciprocal of a fraction is obtained by interchanging the numerator and the denominator.

11. Is the cosecant function defined for all angles?

No, the cosecant function is not defined for angles where the sine function is zero (e.g., 0, π, 2π).

12. How can I find the values of trigonometric functions for other angles?

To find the values of trigonometric functions for other angles, you can use the properties of the unit circle, trigonometric identities, and the ratios of sides in right triangles.

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