TRIGONOMETRYTrigonometryMathematics Calculator
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Cofunction Identities

Cofunctions swap for complementary angles: sin(90°-θ)=cos(θ), cos(90°-θ)=sin(θ), tan(90°-θ)=cot(θ). Complementary angles sum to 90°.

Concept Fundamentals
sin(90°-θ)=cos(θ)
sin↔cos
tan(90°-θ)=cot(θ)
tan↔cot
sec(90°-θ)=csc(θ)
sec↔csc
θ + (90°-θ) = 90°
Complement

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sin(θ)=cos(90°-θ) and cos(θ)=sin(90°-θ) — sine and cosine are cofunctions. Use cofunctions to find exact values: sin(75°)=cos(15°) — sometimes one is easier. In radians: sin(π/2 - θ) = cos(θ). The π/2 is 90° in radian measure.

Key quantities
sin(90°-θ)=cos(θ)
sin↔cos
Key relation
tan(90°-θ)=cot(θ)
tan↔cot
Key relation
sec(90°-θ)=csc(θ)
sec↔csc
Key relation
θ + (90°-θ) = 90°
Complement
Key relation

Ready to run the numbers?

Why: Cofunction identities let you convert between trig functions for complementary angles. sin(30°)=cos(60°) because 30°+60°=90°.

How: For complementary angles α and β (α+β=90°), sin(α)=cos(β), tan(α)=cot(β), sec(α)=csc(β). The 'co' means complement — the functions swap for 90°-θ.

sin(θ)=cos(90°-θ) and cos(θ)=sin(90°-θ) — sine and cosine are cofunctions.Use cofunctions to find exact values: sin(75°)=cos(15°) — sometimes one is easier.
Sources:Wolfram MathWorld — CofunctionKhan Academy — Trig Identities

Run the calculator when you are ready.

Start CalculatingEnter an angle to verify cofunction identities

Examples — Click to Load

cofunction.sh
CALCULATED
$ cofunction --pair sin-cos --angle 30
sin(θ)
0.5
cos(90°-θ)
0.5
Complement
60°
Verified
✓
sin(θ)
0.5
cos(θ)
0.8660254
θ (deg)
30°
90°-θ
60°
Share:
Cofunction Result
sin(θ) = cos(90°-θ)
0.5
θ = 30°90°-θ = 60°Verified ✓
numbervibe.com/calculators/mathematics/trigonometry/cofunction-calculator

Value Breakdown

Function vs Cofunction

Value Split

Calculation Breakdown

INPUT
Input Angle θ
30°
COMPLEMENTARY
Complement 90°-θ
60°
\text{theta} + (90^{circ}-\text{theta} ) = 90^{circ}
IDENTITY
Function Pair
sin(θ) ↔ cos(90°-θ)
ext{Cofunction} ext{identity}
PRIMARY RESULT
sin(θ)
0.5
Direct evaluation
cos(90°-θ)
0.5
Cofunction at complement
VERIFICATION
VERIFIED
✓ Equal
ext{Cofunction} ext{identity} ext{holds}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

↔️

Cofunctions swap for complementary angles — sin and cos, tan and cot, sec and csc.

— Wolfram MathWorld

○

Complementary angles sum to 90° — the two acute angles in a right triangle.

— Khan Academy

Key Takeaways

  • • sin(90°-θ) = cos(θ) and cos(90°-θ) = sin(θ) — complementary angle relationships
  • • tan(90°-θ) = cot(θ) and cot(90°-θ) = tan(θ) — tangent and cotangent are cofunctions
  • • sec(90°-θ) = csc(θ) and csc(90°-θ) = sec(θ) — secant and cosecant are cofunctions
  • • Complementary angles sum to 90° (or π/2 radians) — the two acute angles in a right triangle
  • • On the unit circle, (cos θ, sin θ) for angle θ swaps to (sin θ, cos θ) for 90°-θ — that swap is the cofunction

Did You Know?

📐In a right triangle, the two acute angles are always complementary. sin of one equals cos of the otherSource: Wolfram MathWorld
🔄The unit circle coordinates (cos θ, sin θ) become (sin θ, cos θ) when you rotate by 90° — that swap defines cofunctionsSource: Khan Academy
📜Cofunction identities were known to ancient Greek and Indian mathematicians for computing trig tablesSource: History of Mathematics
∫Calculus: ∫sin x dx = -cos x + C uses the fact that -cos is the antiderivative of sin — a cofunction relationshipSource: Paul's Online Notes
🌅Sunrise/sunset calculations use cofunctions — the elevation angle and zenith angle are complementarySource: Astronomy applications
📱GPS and navigation systems use cofunction identities when converting between coordinate systemsSource: Geodesy

How Cofunction Identities Work

Cofunctions relate an angle to its complement (90° - θ). The complement is the "other" acute angle in a right triangle. Each trig function equals its cofunction evaluated at the complement.

Right Triangle Proof

In a right triangle with angles θ and 90°-θ, sin(θ) = opposite/hypotenuse. For the other angle 90°-θ, the "opposite" becomes the "adjacent" of θ. So sin(90°-θ) = adjacent/hypotenuse = cos(θ).

Unit Circle View

At angle θ, the point is (cos θ, sin θ). At 90°-θ (measured from the y-axis), the point is (sin θ, cos θ). The x and y coordinates swap — hence sin(90°-θ)=cos(θ) and cos(90°-θ)=sin(θ). Use the Unit Circle Calculator to visualize.

Don't Confuse With Supplementary

Complementary angles sum to 90°. Supplementary angles sum to 180°. sin(180°-θ)=sin(θ) is a different identity (supplementary), not cofunction. The Trig Identities Calculator shows both.

Expert Tips

Convert Without Calculator

cos(75°) = sin(15°) = sin(90°-75°). If you know sin(15°)=0.259, you instantly have cos(75°). Try the Sine Calculator.

At 45° They Match

sin(45°)=cos(45°) because 45° is its own complement. Similarly tan(45°)=cot(45°)=1. The Cosine Calculator confirms.

Radians: π/2 - θ

In radians, the complement is π/2 - θ. sin(π/2 - θ) = cos(θ). Same identities, different units.

Calculus Connection

d/dx[sin x] = cos x and d/dx[cos x] = -sin x. The derivative of sin is cos — a cofunction relationship that appears in integration.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorScientific CalcManual
All 3 cofunction pairs✅❌✅
Complement shown✅❌✅
Verification check✅❌✅
Step-by-step breakdown✅❌⚠️
Visual charts✅❌❌
Degrees and radians✅✅✅
Copy & share✅❌❌
Preset examples✅❌❌

Frequently Asked Questions

What is a cofunction?

Cofunctions are trig function pairs where each equals the other evaluated at the complementary angle. sin and cos are cofunctions: sin(90°-θ)=cos(θ). So are tan/cot and sec/csc.

Why do cofunctions work?

In a right triangle, the two acute angles sum to 90°. The sine of one angle equals the cosine of the other because they share the same sides — the opposite of one is the adjacent of the other.

What is the difference between complementary and supplementary?

Complementary angles sum to 90° (e.g. 30° and 60°). Supplementary sum to 180° (e.g. 30° and 150°). Cofunctions use complementary angles.

Does this work in radians?

Yes. The complement of θ in radians is π/2 - θ. sin(π/2 - θ) = cos(θ), etc.

When is sin(θ) = cos(θ)?

When θ = 45° (or π/4). At 45°, the angle equals its complement, so sin(45°)=cos(45°)=√2/2.

Where are cofunctions used?

Simplifying expressions, verifying identities, converting between function forms, calculus (antiderivatives), and right triangle problems.

What about sec and csc?

sec(90°-θ)=csc(θ) because sec=1/cos and csc=1/sin. Since cos(90°-θ)=sin(θ), we get 1/sin(θ)=csc(θ) at the complement.

Can I use cofunctions for obtuse angles?

Yes. The identity holds for all angles. For θ=120°, the complement is -30° (or 330°). cos(120°)=sin(-30°)=-1/2. Both equal -0.5.

Cofunction by the Numbers

3
Cofunction Pairs
90°
Complement Sum
6
Trig Functions
45°
Self-Complement

Disclaimer: Results use IEEE 754 floating-point arithmetic. Cofunction identities hold exactly; small rounding differences may appear. Not a substitute for professional mathematical verification.

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👋Cofunction identities let you convert between trig functions for complementary angles.
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