Calculadora de Identidades Trigonométricas
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
sen²θ + cos²θ = 1 ; tan θ = senθ/cosθ ; sec = 1/cos ; csc = 1/sen
About this calculator
This trigonometric identities calculator takes the value of sin(θ) and optionally the quadrant of the angle to determine the other trigonometric functions: cosine, tangent, cotangent, secant, and cosecant. It uses the fundamental identity sin²θ + cos²θ = 1 to find cosine, and from that calculates the other functions. The quadrant is necessary to define the correct sign of cosine (positive in the first and fourth quadrants; negative in the second and third).
The operation is simple: enter the value of sin(θ) (between -1 and 1) and, if known, select the quadrant (1 to 4). The calculator then computes cos(θ) using √(1 - sin²θ) and applies the sign according to the quadrant. Then it calculates tan(θ) = sinθ/cosθ, cot = 1/tanθ, sec = 1/cosθ, and csc = 1/sinθ. If the quadrant is not provided, cosine is assumed positive (as if the angle is in the first quadrant), which may not be correct in all cases.
You can use this tool in situations such as: solving trigonometry problems that provide only sine, checking homework exercises, or in engineering and physics calculations where only one trigonometric function is known. It is useful for students, teachers, and professionals needing speed and accuracy when working with trigonometric identities.
Cautions: the sine value must be between -1 and 1, otherwise the result will be invalid. Remember that tangent and secant are undefined when cosθ = 0 (angles of 90° and 270°). Similarly, cotangent and cosecant are undefined when sinθ = 0 (0° and 180°). The calculator will display 'undefined' in these cases. Additionally, if the quadrant is not specified, the sign of cosine may be incorrect, affecting all derived functions.
Frequently asked questions
What happens if I don't provide the quadrant?
If the quadrant is not provided, the calculator assumes cosine is positive, as if the angle is in the first quadrant. This may lead to incorrect signs for other functions, so it is recommended to provide the quadrant whenever possible.
Why does the calculator show 'undefined' for some functions?
This occurs when the denominator of the function is zero. For example, tangent and secant are undefined when cosθ = 0 (angles of 90° and 270°). Cotangent and cosecant are undefined when sinθ = 0 (0° and 180°).
Can I use sine values outside the range -1 to 1?
No. The sine of a real angle is always between -1 and 1. Entering a value outside this range will result in an error, as the square root of a negative number is not a real number.
How does the calculator determine the sign of cosine in each quadrant?
The sign of cosine is positive in the first and fourth quadrants, and negative in the second and third. The calculator applies this rule based on the selected quadrant. If no quadrant is chosen, it assumes positive.
Does this calculator work for angles in radians or degrees?
The calculator works regardless of the unit, as it only deals with the numeric values of the functions. You can enter the sine of any angle, whether in degrees or radians, as long as the value is between -1 and 1.