Understanding Cosine
Trigonometry relates angles to side ratios in triangles, and by extension to circular and periodic motion. These relationships appear in physics, engineering, navigation, and wave analysis.
The core relationships
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent = sin/cos
The Pythagorean identity sin²(θ) + cos²(θ) = 1 connects all three and is used constantly to simplify expressions.
Degrees vs. radians
Check whether the calculator and your problem both use degrees or radians. Most scientific formulas use radians. Conversion: radians = degrees × π/180. A full circle is 2π radians = 360°.
Common applications
- Finding missing sides or angles in right triangles (surveying, construction, navigation)
- Modeling periodic phenomena: sound waves, AC current, tides, and mechanical vibration all follow sinusoidal patterns
- Rotational kinematics in engineering and physics
Frequently Asked Questions
What is the formula for cosine?
In a right triangle, cos(angle) equals the length of the side adjacent to the angle divided by the length of the hypotenuse: cos(angle) = adjacent / hypotenuse. Equivalently, on the unit circle, cos(angle) is the x-coordinate of the point where the angle's terminal ray crosses the circle.
How do I convert degrees to radians for cosine?
Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. A full circle is 360° = 2π radians. This calculator converts automatically based on the unit you select.
Why is cosine sometimes negative?
Cosine is the x-coordinate on the unit circle, so it is negative whenever the angle's terminal ray points left of center — between 90° and 270° (Quadrant II and III). It is positive between −90° and 90° (Quadrant I and IV).
What is the range of cosine?
cos(angle) always falls between −1 and 1 inclusive, since it is a coordinate on the unit circle. The value repeats every 360° (2π radians) because cosine is a periodic function.