How to find the exact value of sine 11pi/6?

Introduction

When it comes to trigonometry, finding the exact value of certain angles can be a bit challenging. One such angle is 11π/6, which represents 11 times the angle π/6. In this article, we will explore how to find the exact value of sine 11π/6 and provide answers to related frequently asked questions.

Finding the Exact Value of Sine 11π/6

To find the exact value of sine 11π/6, we need to convert the angle to a reference angle within the unit circle. The unit circle is a circle with a radius of 1 unit, where each point on the circle represents an angle in radians.

The exact value of sine 11π/6 is -1/2. To understand why, we can follow these steps:
1. Start by visualizing the unit circle.
2. The angle 11π/6 is equivalent to an angle of π/6, plus 10 full revolutions (2π) and an additional 5π.
3. Since π/6 lies on the unit circle, its coordinates are (√3/2, 1/2).
4. Adding 10 full revolutions does not change the coordinates since the angle remains the same.
5. Finally, adding 5π to the angle shifts the coordinates to the same position as π/6, just in the opposite direction.
6. Therefore, sine 11π/6 has the same value as sine π/6, which is -1/2.

Related FAQs

1. What is the unit circle?

The unit circle is a circle with a radius of 1 unit, used in trigonometry to represent angles in radians.

2. What is the difference between radians and degrees?

Radians and degrees are two different units of measuring angles. Radians are based on the circumference of a circle, while degrees divide a circle into 360 equal parts.

3. How can I convert radians to degrees?

To convert radians to degrees, multiply the value by 180/π.

4. Is sine positive or negative in the second quadrant?

In the second quadrant, sine values are positive.

5. How do I find the reference angle for a given angle?

To find the reference angle, take the absolute value of the given angle and subtract it from a full revolution (2π) or 180 degrees for degrees.

6. How many radians are in a full revolution?

A full revolution is equal to 2π radians.

7. What is the complementary angle?

The complementary angle is the angle that, when added to the given angle, equals a full revolution (2π) or 90 degrees.

8. How can I find the exact value of cosine 11π/6?

To find the exact value of cosine 11π/6, we can use the fact that cosine is the complementary function of sine. Thus, the cosine of 11π/6 is the negative of the sine of 11π/6, which is -1/2.

9. What is the value of sine 7π/4?

Following a similar approach as sine 11π/6, sine 7π/4 has the exact value of -1/√2, or -√2/2.

10. How can I determine if an angle is in the first or second quadrant?

Angles between 0 and π/2 radians lie in the first quadrant, while angles between π/2 and π radians are in the second quadrant.

11. Is there a difference between sine and arcsine?

Yes, there is a difference. Sine is a trigonometric function used to find the ratio of the length of a triangle’s side to its hypotenuse, whereas arcsine (or inverse sine) is used to find the angle when the ratio is known.

12. Can the sine of an angle exceed 1 or be below -1?

No, the sine function always returns a value between -1 and 1, inclusive. If you encounter a value outside this range, it may be due to an error.

Dive into the world of luxury with this video!