cos θ = ±√(1 - sin² θ)
Cosine is calculated using the fundamental trigonometric identity
Enter a decimal number (-1 ≤ sin θ ≤ 1), fraction (1/2), or expression with square root (√2/2)
Choose quadrant or show all solutions:
Sine to Cosine Calculator
This calculator determines the cosine value when you know the sine value of an angle using the fundamental trigonometric identity. It supports various input formats including decimals, fractions, and square root expressions.
cos θ = ±√(1 - sin² θ)
Pythagorean trigonometric identity
Key Features:
- Multiple input formats: Accept decimals (0.5), fractions (1/2), and expressions with square roots (√2/2)
- Quadrant selection: Choose specific quadrants or view all possible solutions
- Visual representation: Interactive unit circle diagram showing angle positions
- Additional calculations: Displays corresponding angles, tangent, and cotangent values
Example Calculation:
Input: sin θ = 0.6
Calculation: cos θ = ±√(1 - 0.6²) = ±√(1 - 0.36) = ±√0.64 = ±0.8
Results:
- Positive solution: cos θ = 0.8 (Quadrants I and IV)
- Negative solution: cos θ = -0.8 (Quadrants II and III)
The calculator automatically validates that input sine values are within the valid range of -1 to 1, as required by the mathematical definition of sine function. When you select "All solutions", both positive and negative cosine values are displayed with their corresponding quadrants and angle measures.
Important: Since cosine can be positive or negative depending on the quadrant, this calculator shows both possible values unless you specify a particular quadrant. The sign of cosine is positive in Quadrants I and IV, and negative in Quadrants II and III.
Perfect for students, engineers, and professionals working with trigonometric calculations who need to find cosine values from known sine values in various mathematical contexts.