Finding the exact value of cosine for any given angle can be a challenging task, especially when dealing with non-standard angles. In this article, we will explore a step-by-step approach to determining the exact value of cosine for the angle 135 degrees.
Understanding Cosine
Before we dive into the process of finding the exact value of cosine for 135 degrees, it is essential to understand what cosine represents. Cosine is a trigonometric function that relates the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle.
To visualize this, consider a right-angled triangle with one angle measuring 135 degrees. The adjacent side of this angle is the side that is closest to it, while the hypotenuse is the longest side of the triangle. By determining the ratio of these two sides, we can find the cosine of the angle.
Finding the Exact Value of Cosine of 135 Degrees
To find the exact value of cosine for 135 degrees, we can utilize the periodicity of the cosine function. Cosine has a period of 360 degrees, meaning the cosine of an angle repeats after adding or subtracting multiples of 360 degrees.
Therefore, we can convert the given angle, 135 degrees, into an equivalent angle within one period of the cosine function. By subtracting 360 degrees from 135 degrees, we obtain an equivalent angle of -225 degrees. The cosine of -225 degrees is equal to the cosine of the original angle, 135 degrees.
Now, let’s find the exact value of the cosine of -225 degrees. We can use the unit circle, a circle with a radius of 1 unit, to evaluate the cosine function.
Step 1:
Draw a circle and mark the point where the circle intersects the positive x-axis as the origin.
Step 2:
Divide the circle into four equal quadrants by drawing two perpendicular lines, one passing through the origin and the other through the top of the circle.
Step 3:
Since we are interested in the cosine value, focus on the x-coordinate of the unit circle at -225 degrees.
Step 4:
Draw a line from the origin to the point on the unit circle at -225 degrees.
Step 5:
Determine the length of the x-coordinate (the adjacent side) where the line intersects the unit circle.
Step 6:
The length of the x-coordinate is the exact value of cosine for the angle -225 degrees.
The exact value of the cosine of 135 degrees is -√2/2.
Related FAQs:
1. What is the cosine of 45 degrees?
The cosine of 45 degrees is equal to √2/2.
2. How can I find the cosine of an obtuse angle?
To find the cosine of an obtuse angle, follow the same steps as mentioned above but ensure the angle falls within one period of the cosine function (0 to 360 degrees).
3. What is the cosine of 180 degrees?
The cosine of 180 degrees is -1.
4. Can cosine be greater than 1 or less than -1?
No, the cosine function ranges between -1 and 1, inclusive.
5. How does cosine relate to the unit circle?
Cosine represents the x-coordinate of a point on the unit circle corresponding to a specific angle.
6. What is the cosine of 0 degrees?
The cosine of 0 degrees is 1.
7. What is the cosine of 90 degrees?
The cosine of 90 degrees is 0.
8. How does cosine relate to sine?
Cosine and sine are related through the Pythagorean identity: sin^2θ + cos^2θ = 1.
9. What is the cosine of 270 degrees?
The cosine of 270 degrees is 0.
10. How can I find the cosine of a non-standard angle?
To find the cosine of a non-standard angle, convert it to an equivalent angle within one period of the cosine function and then evaluate it using the unit circle or other trigonometric identities.
11. What is the cosine of 60 degrees?
The cosine of 60 degrees is 1/2.
12. Are there any special angles to remember for cosine?
Yes, the values of cosine for 0, 30, 45, 60, and 90 degrees are easy to remember and frequently used in trigonometric calculations.