Cofunction Calculator – Guide & Formulas
Calculate trigonometric cofunctions instantly. Find sin↔cos, tan↔cot, sec↔csc pairs and verify cofunction identities.
The Cofunction Calculator demonstrates the relationship between trigonometric cofunctions of complementary angles. Enter an angle to see how sin(θ) = cos(90° - θ), tan(θ) = cot(90° - θ), and sec(θ) = csc(90° - θ) work in practice.
Key Takeaway
Use the free Cofunction Calculator to calculate trigonometric cofunctions instantly. find sin↔cos, tan↔cot, sec↔csc pairs and verify cofunction identities. Get instant results with step-by-step explanations.
How to Use the Cofunction Calculator
- Enter an angle in degrees.
- Select a trigonometric function.
- View the cofunction result and the complementary angle.
- See the cofunction identity verified with actual values.
The Formula
Variable Definitions
- θ: The input angle in degrees
- 90° - θ: The complementary angle
- sin/cos: Sine and cosine are cofunctions
- tan/cot: Tangent and cotangent are cofunctions
- sec/csc: Secant and cosecant are cofunctions
Example: Cofunction of sin(30°)
Verify that sin(30°) = cos(60°).
- Step 1: Compute sin(30°) = 0.5.
- Step 2: Find the complementary angle: 90° - 30° = 60°.
- Step 3: Compute cos(60°) = 0.5.
- Step 4: Verify: sin(30°) = cos(60°) = 0.5 ✓. The cofunction identity holds.
Frequently Asked Questions
What are cofunctions?
Cofunctions are pairs of trigonometric functions that are equal when evaluated at complementary angles. The pairs are: sine and cosine, tangent and cotangent, secant and cosecant.
What is the cofunction identity?
The cofunction identities state that sin(θ) = cos(90° - θ), tan(θ) = cot(90° - θ), and sec(θ) = csc(90° - θ). These identities arise from the relationship between an angle and its complement in a right triangle.
What are complementary angles?
Two angles are complementary if they add up to 90°. For example, 30° and 60° are complementary because 30° + 60° = 90°. The cofunction identity relates a function of an angle to the cofunction of its complement.
Why are sine and cosine cofunctions?
In a right triangle, sin(θ) equals the ratio of the opposite side to the hypotenuse, while cos(θ) equals the ratio of the adjacent side to the hypotenuse. Since the opposite side of one acute angle is the adjacent side of the other, sin(θ) = cos(90° - θ).
What is the cofunction of tan?
The cofunction of tangent is cotangent (cot). The identity is tan(θ) = cot(90° - θ). For example, tan(30°) = cot(60°) = 1/√3 ≈ 0.577.
What is the cofunction of sec?
The cofunction of secant is cosecant (csc). The identity is sec(θ) = csc(90° - θ). For example, sec(30°) = csc(60°) = 2/√3 ≈ 1.155.
Can cofunction identities be used to simplify expressions?
Yes. Cofunction identities are frequently used to simplify trigonometric expressions, solve equations, and prove other identities. For example, sin(π/2 - x) can be replaced with cos(x).
How do cofunction identities relate to the unit circle?
On the unit circle, sin(θ) equals the y-coordinate and cos(θ) equals the x-coordinate. The cofunction identity sin(θ) = cos(90° - θ) reflects the symmetry between the x and y axes when the angle is measured from different reference axes.
What is the cofunction theorem?
The cofunction theorem states that if two acute angles are complementary, then the sine of one equals the cosine of the other, and similarly for tangent/cotangent and secant/cosecant. This is a fundamental property of right triangle trigonometry.
Are cofunction identities used in calculus?
Yes. Cofunction identities appear in integration and differentiation, especially when converting between trigonometric forms. For example, ∫sin(x)dx = -cos(x) + C relates sine and cosine through a cofunction-like relationship.