When working with trigonometry, finding the exact value of trigonometric functions can sometimes be challenging. One common question that students come across is how to find the exact value of sine 120 degrees. In this article, we will explore different approaches to solve this problem and provide a step-by-step explanation.
Understanding Sine Function
Before we jump into finding the value of sine 120 degrees, let’s briefly understand the sine function. Sine is one of the primary trigonometric functions and is denoted as “sin.” It relates the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
In a unit circle, where the radius is equal to 1, the sine of an angle is represented by the y-coordinate of the point where the terminal side intersects the unit circle. The sine function is periodic, meaning it repeats itself after every 360 degrees or 2π radians.
So, let’s dive into solving the question at hand.
Finding the Exact Value of Sine 120 Degrees
To find the exact value of sine 120 degrees, we can use a couple of different methods. One approach is to use the trigonometric identity for sine of a supplementary angle. The other approach involves using the values of commonly known angles.
Method 1: Using the Trigonometric Identity
The trigonometric identity for sine of a supplementary angle states that sine of an angle and its supplement are equal but have opposite signs. The supplement of 120 degrees is 60 degrees, which belongs to the family of angles (30, 60, 90).
Therefore, we can write:
**sin(120 degrees) = -sin(60 degrees)**
The sine of 60 degrees is a well-known value, which is √3/2. Therefore:
sin(120 degrees) = -√3/2
So, the exact value of sine 120 degrees is **-√3/2**.
Method 2: Using the Known Angle Values
Another method to find the exact value is by utilizing the values of commonly known angles: 0, 30, 45, 60, and 90 degrees.
Since sine is periodic and repeats after every 360 degrees, we can subtract 120 degrees from 180 degrees to find an equivalent angle:
180 degrees – 120 degrees = 60 degrees
We already know the value of sine 60 degrees, which is √3/2. However, this time the negative sign will be introduced as the angle is in the second quadrant.
Therefore, sin(120 degrees) = -√3/2
Thus, both methods lead us to the same result.
Frequently Asked Questions:
1. How do you find the exact value of sin 150 degrees?
Using the same methods explained above, the exact value of sin 150 degrees can be found as -(√3/2).
2. How can I calculate sine using a calculator?
Most scientific calculators have a dedicated “sin” button that allows you to calculate the sine of any given angle.
3. Is there a shortcut to memorize trigonometric values?
Yes, there are many mnemonic devices and patterns that can help you remember the values of commonly used angles in trigonometry.
4. What’s the value of sin 0 degrees?
The sine of 0 degrees is 0.
5. What is the range of the sine function?
The range of the sine function is -1 to 1.
6. How do you prove the value of sin 30 degrees?
By referring to the unit circle or using the trigonometric identity, you can prove that sin 30 degrees is equal to 1/2.
7. Is it possible to find the exact value of sin 135 degrees?
Yes, using the methods explained earlier, the exact value of sin 135 degrees can be found as -√2/2.
8. Can the value of sine be greater than 1?
No, the value of sine is always between -1 and 1, inclusive.
9. How does the value of sine change in different quadrants?
In the first and second quadrants, the sine function is positive, while in the third and fourth quadrants, it is negative.
10. How do you find the sine of a negative angle?
The sine of a negative angle is equal to the negative of the sine of the corresponding positive angle.
11. What is the difference between sin and sinh functions?
The sin function deals with angles in trigonometry, while the sinh function deals with exponential values in mathematics.
12. Can the exact value of sine be irrational?
Yes, many angles in trigonometry yield irrational values for sine, such as sin 45 degrees equals √2/2 and sin 30 degrees equals 1/2.
By understanding the concepts and utilizing the trigonometric identities, finding the exact value of sine for various angles becomes easier. Remembering the values of common angles and using them as a reference can also aid in the process.