TRIGONOMETRYTrigonometryMathematics Calculator
cos

The Cosine Function

Cosine relates an angle to the ratio of the adjacent side over the hypotenuse. On the unit circle, cos(θ) is the x-coordinate — the horizontal component of the point at angle θ.

Concept Fundamentals
[-1, 1]
Range
2π (360°)
Period
cos(-θ) = cos(θ)
Even Function
cos(θ) = sin(90°-θ)
Cofunction

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Cosine is positive in quadrants 1 and 4 (where x > 0), negative in 2 and 3. Remember C in ASTC. cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2, cos(90°)=0 — the pattern mirrors sine. Cosine similarity in machine learning measures the angle between vectors using cos(θ).

Key quantities
[-1, 1]
Range
Key relation
2π (360°)
Period
Key relation
cos(-θ) = cos(θ)
Even Function
Key relation
cos(θ) = sin(90°-θ)
Cofunction
Key relation

Ready to run the numbers?

Why: Cosine models horizontal displacement and in-phase oscillations. It is essential for signal processing, Fourier analysis, and structural engineering.

How: cos(θ) = adjacent/hypotenuse in a right triangle. On the unit circle, it equals the x-coordinate. Cosine is sine shifted by 90°: cos(θ) = sin(θ + 90°).

Cosine is positive in quadrants 1 and 4 (where x > 0), negative in 2 and 3. Remember C in ASTC.cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2, cos(90°)=0 — the pattern mirrors sine.
Sources:Wolfram MathWorld — CosineKhan Academy — Trigonometry

Run the calculator when you are ready.

Start CalculatingEnter an angle to compute cos(θ) and all related trig values

Examples — Click to Load

cosine.sh
CALCULATED
$ cos --angle 45° --all-functions
cos(θ)
0.70710678
sin(θ)
0.70710678
tan(θ)
1
Quadrant
Q1
csc(θ)
1.41421356
sec(θ)
1.41421356
cot(θ)
1
Ref Angle
45°
Share:
Cosine Calculator Result
cos(45°)
0.70710678
Q1ref 45°sin²+cos² = 1
numbervibe.com/calculators/mathematics/trigonometry/cosine-calculator

Trig Value Breakdown

All 6 Trig Functions

cos² vs sin² (Pythagorean Identity)

Calculation Breakdown

CONVERSION
Input Angle
45°
Convert to Radians
0.78539816 rad
45° × π/180
Normalized Angle
45°
\text{theta} mod 360^{circ}
Quadrant
Q1
45.0° is in quadrant 1
Reference Angle
45°
ext{Acute} ext{angle} ext{to} x- ext{axis}
PRIMARY RESULT
COSINE VALUE
0.70710678
cos(45°)
RELATED VALUES
Sine
0.70710678
sin(45°)
Tangent
1
tan(45°) = sin/cos
Cosecant
1.41421356
1/\text{sin}(\text{theta} )
Secant
1.41421356
1/\text{cos}(\text{theta} )
Cotangent
1
\text{cos}(\text{theta} )/\text{sin}(\text{theta} )
PYTHAGOREAN IDENTITY
cos²(θ)
0.5
sin²(θ)
0.5
sin²(θ) + cos²(θ)
1
ext{Always} ext{equals} 1

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

🧮 Fascinating Math Facts

📐

Cosine gets its name from 'complement of sine' — cos(θ) = sin(90° - θ).

— Wolfram MathWorld

↔️

Cosine is even: cos(-θ) = cos(θ). The graph is symmetric about the y-axis.

— Paul's Online Notes

Key Takeaways

  • • cos(θ) = adjacent / hypotenuse in a right triangle; on the unit circle it is the x-coordinate
  • • Range is always [-1, 1]. Key values: cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(90°) = 0, cos(180°) = -1
  • • Cosine is an even function: cos(-θ) = cos(θ) with period 2π (360°)
  • • The Pythagorean identity sin²(θ) + cos²(θ) = 1 always holds
  • • Cosine is positive in Q1 and Q4 (0°–90° and 270°–360°), negative in Q2 and Q3 (90°–270°)

Did You Know?

📐Cosine gets its name from 'complement of sine' — cos(θ) = sin(90° - θ). The cofunction identity links cosine to sine for complementary anglesSource: Wolfram MathWorld
↔️Cosine represents horizontal displacement on the unit circle — it's the x-coordinate. Sine is vertical (y). Together they trace the circleSource: Khan Academy
🔄Cosine is even: cos(-θ) = cos(θ). The graph is symmetric about the y-axis, unlike sine which is odd and symmetric about the originSource: Paul's Online Notes
📡In signal processing and Fourier analysis, cosine terms represent the "in-phase" component of a signal — critical for AC circuits and radioSource: MIT OpenCourseWare
🏛️Architects use cosine to calculate horizontal projections of inclined beams and roof slopes — the adjacent side in structural trianglesSource: Engineering Toolbox
🎯Cosine similarity is a fundamental metric in machine learning and NLP for comparing vectors — it measures the angle between themSource: IEEE Machine Learning

How the Cosine Function Works

The cosine function relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. On the unit circle, it is simply the x-coordinate of the point at angle θ.

Unit Circle Definition

For any angle θ measured counterclockwise from the positive x-axis, the terminal point on a circle of radius 1 has coordinates (cos θ, sin θ). Cosine is the horizontal component — it starts at 1 when θ = 0° and returns to 1 after a full rotation.

Even Function Symmetry

Cosine is even: cos(-θ) = cos(θ). Reflecting the angle across the x-axis leaves cosine unchanged. This contrasts with sine, which is odd. The cosine graph is symmetric about the y-axis.

Cofunction Identity

cos(θ) = sin(90° - θ) and sin(θ) = cos(90° - θ). For complementary angles, sine and cosine swap. So cos(60°) = sin(30°) = 0.5. This identity is essential for solving many trig problems.

Expert Tips

Memorize the Special Angles

cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2, cos(90°)=0. Cosine "starts high" at 0° — the pattern mirrors sine but shifted. Use the Unit Circle Calculator to visualize.

Q1 and Q4 Are Positive

Cosine is positive where x > 0: Quadrants 1 and 4. Negative in Q2 and Q3. Remember "All Students Take Calculus" — Cosine is positive in the 4th quadrant. Try the Sum & Difference Calculator.

Phase Shift from Sine

cos(θ) = sin(θ + 90°). Cosine is sine shifted left by 90°. In wave analysis, this 90° phase difference is why AC voltage and current are often "out of phase" in inductors and capacitors.

The Pythagorean Identity

sin²θ + cos²θ = 1 always. If you know cos(θ), then sin(θ) = ±√(1 - cos²θ), with the sign from the quadrant. See the Trig Identities Calculator.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorScientific CalculatorManual Computation
All 6 trig functions at once✅❌ One at a time❌
Quadrant & reference angle✅❌✅ Slow
Visual charts & breakdown✅❌❌
Step-by-step explanation✅❌✅
Pythagorean identity check✅❌⚠️ Manual
Copy & share results✅❌❌
Degrees and radians✅✅✅
Preset examples✅❌❌

Frequently Asked Questions

What is the range of the cosine function?

The cosine function always outputs values between -1 and 1, inclusive. cos(0°) = 1 is the maximum and cos(180°) = -1 is the minimum. Like sine, this bounded range makes cosine ideal for modeling oscillations and circular motion.

Why is cosine called an even function?

A function is even when f(-x) = f(x). For cosine: cos(-θ) = cos(θ). Graphically, the cosine curve is symmetric about the y-axis — reflecting it produces the same graph. This contrasts with sine, which is odd: sin(-θ) = -sin(θ).

Where is cosine positive and negative?

Cosine is positive in Quadrants 1 and 4 (where x > 0 on the unit circle): 0°–90° and 270°–360°. It is negative in Quadrants 2 and 3 (90°–270°) where x < 0. Remember: cosine = x-coordinate.

What is the cofunction identity for cosine?

cos(θ) = sin(90° - θ) and sin(θ) = cos(90° - θ). For complementary angles that sum to 90°, sine and cosine swap. So cos(20°) = sin(70°). This identity is fundamental in trigonometry.

How do I find cosine without a calculator?

Memorize the special angles: cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2≈0.707, cos(60°)=1/2, cos(90°)=0. For other angles, use reference angles and quadrant signs, or the Taylor series: cos(x) ≈ 1 - x²/2 + x⁴/24.

How is cosine related to sine?

They are cofunctions and phase-shifted: cos(θ) = sin(θ + 90°). The Pythagorean identity sin²θ + cos²θ = 1 links them. On the unit circle, (cos θ, sin θ) gives the coordinates of the point at angle θ.

What is the period of cosine?

The period of cos(θ) is 2π radians (360°), meaning cos(θ + 2π) = cos(θ). For cos(Bθ), the period becomes 2π/B. Cosine and sine share the same period.

Where is cosine used in real life?

Cosine appears in physics (wave motion, projections), engineering (structural analysis, signal processing), computer graphics (rotations, 3D transforms), navigation (bearing calculations), and machine learning (cosine similarity).

Cosine Function by the Numbers

[-1, 1]
Output Range
360°
Period
Even
Symmetry
x-axis
Unit Circle

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (aerospace, medical devices), always verify with certified computational tools. Not a substitute for professional engineering analysis.

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