Cosine Calculator

How to Calculate or Solve Cosine (cos)

The cosine (cos) of an angle is a ratio used in trigonometry to relate the sides of a right-angled triangle. It can also be calculated directly using a calculator or reference tables if the angle is known.

1. Cosine in a Right-Angled Triangle

In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

Formula:
cos(θ) = Adjacent / Hypotenuse

Steps:

1. Identify the angle θ in the triangle.
2. Find the length of the side adjacent to the angle.
3. Find the length of the hypotenuse (the longest side).
4. Divide the adjacent side by the hypotenuse.

Example:

If the adjacent side is 4 units and the hypotenuse is 5 units:

cos(θ) = 4 / 5 = 0.8

2. Cosine of a Known Angle

You can find the cosine of common angles using a calculator or trigonometric table.

Common Cosine Values:

• cos(0°) = 1
• cos(30°) = √3/2 ≈ 0.866
• cos(45°) = √2/2 ≈ 0.707
• cos(60°) = 0.5
• cos(90°) = 0
• cos(180°) = -1

3. Degrees vs Radians

Angles can be measured in degrees (°) or radians. When solving cosine for an angle in radians, you may use expressions like π (pi) to represent angles.

Conversion Formula:

Radians = Degrees × (π / 180)

Examples:

• 90° = π/2 radians
• 60° = π/3 radians
• 45° = π/4 radians

4. Using π (Pi) in Cosine Calculations

π (pi) is a mathematical constant used to express angles in radians. For example:

• cos(π/2) = 0
• cos(π) = -1
• cos(2π) = 1

Key Points to Remember:

• Cosine is a ratio in a right triangle: adjacent ÷ hypotenuse.
• Cosine values always fall between -1 and 1.
• You can use either degrees or radians, but be consistent.
• Use the π symbol to work easily with radians (e.g., π/2, π/3).