0.5
0.5
-1.6653345369e-16
0.8660254
0.8660254
1.1102230246e-16
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0
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0
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0
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0.5
0.5
-1.6653345369e-16
0.8660254
0.8660254
1.1102230246e-16
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0
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0
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0
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0
The Cofunction Calculator evaluates the cofunction identities of trigonometry, showing that each trigonometric function of an angle equals its cofunction of the complementary angle. These identities are among the most elegant relationships in trigonometry and form the foundation for simplifying expressions and solving equations.
Cofunction identities establish a deep connection between pairs of trigonometric functions. Two angles are complementary when they sum to 90° (or $$\frac{\pi}{2}$$ radians). The six cofunction identities are:
$$\sin(\theta) = \cos(90° - \theta)$$
$$\cos(\theta) = \sin(90° - \theta)$$
$$\tan(\theta) = \cot(90° - \theta)$$
$$\cot(\theta) = \tan(90° - \theta)$$
$$\sec(\theta) = \csc(90° - \theta)$$
$$\csc(\theta) = \sec(90° - \theta)$$
The word "cosine" literally means "complement's sine." In a right triangle with acute angles $$\theta$$ and $$90° - \theta$$, the side that is adjacent to $$\theta$$ is opposite to $$90° - \theta$$. This geometric fact means that:
$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \cos(90° - \theta)$$
This relationship extends beyond the first quadrant through the unit circle definition of trigonometric functions.
On the unit circle, the point at angle $$\theta$$ has coordinates $$(\cos\theta, \sin\theta)$$. The point at angle $$90° - \theta$$ has coordinates $$(\cos(90° - \theta), \sin(90° - \theta))$$. The cofunction identities tell us that swapping the angle with its complement swaps the x and y coordinates.
Cofunction identities are essential tools in calculus, particularly when evaluating integrals. The substitution $$u = \frac{\pi}{2} - x$$ transforms sine integrals into cosine integrals and vice versa. They also simplify trigonometric equations by converting mixed-function expressions into single-function forms.
In physics, cofunction identities appear naturally when decomposing forces and velocities into components. A force at angle $$\theta$$ from the horizontal has a vertical component involving $$\sin\theta$$, which equals $$\cos(90° - \theta)$$—the horizontal component at the complementary angle.
Enter any angle in degrees. The calculator displays all six trigonometric functions alongside their cofunctions evaluated at the complementary angle, confirming the cofunction identities numerically. Each pair of values should match exactly (within floating-point precision).
The calculator converts the input angle $$\theta$$ and its complement $$90° - \theta$$ to radians, then evaluates all six trigonometric functions for both angles. The cofunction pairs are displayed side by side so you can verify that $$\sin(\theta) = \cos(90° - \theta)$$ and all other identities hold.
Each pair of outputs should show identical (or nearly identical) values. If $$\sin(30°) = 0.5$$, then $$\cos(60°)$$ should also equal $$0.5$$. Any tiny differences are due to floating-point arithmetic, not errors in the identities. Values showing NaN indicate the function is undefined at that angle (e.g., $$\tan(90°)$$).
Inputs
Results
sin(30°) = cos(60°) = 0.5 and cos(30°) = sin(60°) ≈ 0.866, confirming the cofunction identities for a well-known angle.
Inputs
Results
For θ = 50°, the complement is 40°. Every cofunction pair matches, verifying sin(50°) = cos(40°), tan(50°) = cot(40°), and sec(50°) = csc(40°).
Cofunction identities state that the value of a trigonometric function at an angle equals the value of its cofunction at the complementary angle (90° minus the original angle). The three cofunction pairs are sine-cosine, tangent-cotangent, and secant-cosecant.
The prefix "co-" in cosine stands for "complement." Cosine literally means "complement's sine," reflecting the identity $$\cos(\theta) = \sin(90° - \theta)$$. Similarly, cotangent is the complement's tangent and cosecant is the complement's secant.
Yes. Although cofunctions are most intuitive for acute angles in a right triangle, the identities hold for all real-valued angles when trigonometric functions are defined via the unit circle. The complement $$90° - \theta$$ can be negative or greater than 90°.
They are used to simplify integrals and derivatives. For example, the integral $$\int_0^{\pi/2} \sin^n(x)\,dx = \int_0^{\pi/2} \cos^n(x)\,dx$$ follows directly from the substitution $$u = \pi/2 - x$$ and the cofunction identity.
At $$\theta = 45°$$, the complement is also 45°, so every trigonometric function equals its own cofunction: $$\sin(45°) = \cos(45°)$$, $$\tan(45°) = \cot(45°) = 1$$, and $$\sec(45°) = \csc(45°) = \sqrt{2}$$.
Absolutely. In radians, the identities use $$\frac{\pi}{2}$$ instead of 90°. For example, $$\sin(\theta) = \cos\!\left(\frac{\pi}{2} - \theta\right)$$ and $$\tan(\theta) = \cot\!\left(\frac{\pi}{2} - \theta\right)$$.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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