Cos 135 degrees describes the cosine of a 135-degree angle in standard position on the unit circle. This value is a key reference in trigonometry, especially when analyzing angles beyond the first quadrant.
Understanding cos 135 degrees helps with practical applications in physics, engineering, and computer graphics where directional components must be calculated accurately.
Unit Circle Definition
Coordinates on the Unit Circle
On the unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side intersects the circle. For 135 degrees, this point lies in the second quadrant.
| Angle (degrees) | Quadrant | x (Cosine) | y (Sine) |
|---|---|---|---|
| 135 | Second | -√2/2 | √2/2 |
| 45 | First | √2/2 | √2/2 |
| 225 | Third | -√2/2 | -√2/2 |
| 315 | Fourth | √2/2 | -√2/2 |
Reference Angle Approach
Using 45 Degrees as Reference
The reference angle for 135 degrees is 45 degrees, because 180 minus 135 equals 45. This relationship simplifies computation of trigonometric values.
Since cosine is negative in the second quadrant, cos 135 degrees equals the negative of cos 45 degrees, resulting in -√2/2.
Exact Value and Simplification
Radical and Decimal Forms
The exact value of cos 135 degrees is -√2/2. In decimal form, this is approximately -0.7071067812, which is useful in computational and graphical contexts.
Both representations appear frequently in textbooks, calculators, and software libraries that implement trigonometric functions.
Graph Behavior
Position on the Cosine Curve
On the graph of y = cos x, the point at 135 degrees (or 3π/4 radians) shows a smooth transition from positive to negative values. The function decreases continuously through this region.
Visualizing this segment of the curve reinforces why the output for 135 degrees is negative while the angle remains between 90 and 180 degrees.
Applications in Physics and Engineering
Resolving Forces and Vectors
In mechanics, cos 135 degrees appears when resolving forces acting at 135 degrees from a reference axis. The negative sign indicates direction opposite to the positive x-axis.
Engineers use this value to compute horizontal components in structural analysis, robotics path planning, and signal processing workflows.
Key Takeaways
- Cos 135 degrees equals -√2/2 on the unit circle.
- The reference angle is 45 degrees, located in the second quadrant.
- The negative sign reflects the leftward direction of the x-coordinate.
- This value is widely applied in physics, engineering, and computer graphics.
- Understanding quadrant rules ensures accurate trigonometric computation.
FAQ
Reader questions
Why is cos 135 degrees negative while the angle is between 90 and 180 degrees?
Cosine represents the x-coordinate on the unit circle, and the x-coordinate is negative in the second quadrant, which is why cos 135 degrees yields a negative value.
How does cos 135 degrees relate to cos 45 degrees?
Because 135 degrees and 45 degrees share the same reference angle, their absolute cosine values are equal, but cos 135 degrees is negative due to quadrant placement.
Can cos 135 degrees be used directly in physics simulations?
Yes, the value -√2/2 is directly usable in vector decomposition, wave equations, and other physics simulations where directional accuracy is essential.
What are common tools for verifying the value of cos 135 degrees?
Scientific calculators, programming language math libraries, and digital trigonometry tables all provide consistent results for cos 135 degrees.