Scientific Calculator — Free Online Calculator with Trig, Log, and Scientific Functions

Scientific Calculator Functions

Common Uses by Subject

Degrees vs. Radians

Trigonometric functions can take arguments in either degrees or radians. This is one of the most common sources of error when using a scientific calculator.

• Degrees: Full circle = 360°. Right angle = 90°. Used in most everyday geometry and navigation.
• Radians: Full circle = 2π rad ≈ 6.283 rad. Right angle = π/2 rad ≈ 1.571 rad. Used in calculus, physics, and engineering mathematics.

Conversion: degrees × (π/180) = radians; radians × (180/π) = degrees.

Always check which mode your calculator is in before computing trigonometric values. sin(90) in DEG mode = 1.000 (correct). sin(90) in RAD mode ≈ 0.894 (sin of 90 radians, which is a very different angle). The calculator displays DEG or RAD prominently to prevent this error.

Logarithm Functions

log (base 10)

log₁₀(x) asks: "10 to what power gives x?" — log₁₀(100) = 2, log₁₀(1000) = 3, log₁₀(0.01) = −2. Used in pH calculations, decibels (sound intensity), Richter magnitude (earthquakes), and any scale involving orders of magnitude.

ln (natural logarithm)

ln(x) uses base e ≈ 2.71828. It is the inverse of e^x. Used in continuous compound interest, population growth models, radioactive decay, and throughout calculus and physics. Key values: ln(1) = 0; ln(e) = 1; ln(e²) = 2.

Change of base

To compute log₂(x) on a calculator that only has log₁₀ and ln: use log₂(x) = log₁₀(x) / log₁₀(2) = ln(x) / ln(2). Example: log₂(32) = ln(32)/ln(2) = 3.466/0.693 = 5 (correct — 2⁵ = 32).

Powers and Scientific Notation

Large and small numbers in science are expressed in scientific notation: N × 10ⁿ where 1 ≤ N < 10. Enter these on the calculator using the EXP or EE button:

• 6.022 × 10²³ (Avogadro's number): enter 6.022 EXP 23
• 1.602 × 10⁻¹⁹ (elementary charge): enter 1.602 EXP −19
• 3 × 10⁸ (speed of light): enter 3 EXP 8

The calculator displays results in scientific notation automatically when values exceed display capacity (typically beyond 10⁹ or below 10⁻⁴). The scientific notation converter provides dedicated conversion between standard and scientific notation forms.

Factorial and Combinatorics

Factorial (n!)

n! = n × (n−1) × (n−2) × ... × 2 × 1. Used in permutations and combinations. 5! = 5 × 4 × 3 × 2 × 1 = 120. 0! = 1 by convention (important for combinatorics formulas to work correctly).

Combinations (nCr)

nCr = n! / (r! × (n−r)!) — the number of ways to choose r items from n items without regard to order. Used extensively in probability. 10C3 = 10!/(3! × 7!) = 120. On the calculator: enter n, press nCr, enter r, press =.

Permutations (nPr)

nPr = n! / (n−r)! — the number of ways to choose r items from n items where order matters. 10P3 = 10!/(10−3)! = 720. Permutations count arrangements; combinations count selections.

Using Memory Effectively

The memory functions (M+, M−, MR, MC) are valuable for multi-step calculations:

1. Store an intermediate result: Compute first, press M+ to store in memory
2. Continue the calculation: Perform subsequent steps
3. Recall the stored value: Press MR when you need the stored value
4. Add to memory: M+ adds the current result to whatever is in memory (useful for running totals)
5. Subtract from memory: M− subtracts the current result from memory
6. Clear memory: MC resets memory to zero

Example — computing the quadratic formula for x² − 5x + 6 = 0 (a=1, b=−5, c=6):

1. Compute b² = (−5)² = 25
2. Compute 4ac = 4 × 1 × 6 = 24
3. Discriminant = 25 − 24 = 1; √1 = 1; press M+
4. Numerator for x₁: −b + MR = 5 + 1 = 6; divide by 2a = 2 → x₁ = 3
5. Numerator for x₂: −b − MR = 5 − 1 = 4; divide by 2 → x₂ = 2

Angle Calculations

Inverse trigonometric functions

asin (arcsin), acos (arccos), and atan (arctan) are the inverse functions — they take a ratio and return an angle. Example: in a right triangle where the opposite side = 3 and hypotenuse = 5, sin(θ) = 3/5 = 0.6. To find θ: θ = asin(0.6) = 36.87°.

Range restrictions: asin returns [−90°, 90°]; acos returns [0°, 180°]; atan returns [−90°, 90°]. For angles in other quadrants, adjust using the known quadrant of the triangle or vector.

Common Questions

Why does my calculator give an error for tan(90°)?

tan(90°) is mathematically undefined — it equals sin(90°)/cos(90°) = 1/0, which is undefined. Some calculators show a very large number (approaching infinity) or an error. This is correct — there is no defined answer. tan is undefined at 90° and 270° (and all odd multiples of 90°).

What is the difference between log and ln?

log typically means log base 10 (common logarithm) on most scientific calculators. ln means the natural logarithm (base e ≈ 2.71828). In pure mathematics texts, log sometimes means natural logarithm — check the context. On the calculator: log(100) = 2 (since 10² = 100); ln(100) ≈ 4.605 (since e^4.605 ≈ 100).

Why is e special in mathematics?

Euler's number e ≈ 2.71828 is the base of natural logarithms and is the unique number where the function eˣ is its own derivative. This property makes eˣ appear naturally in any situation involving rates of change: compound interest, population growth, radioactive decay, heat transfer, and countless physics and engineering equations. e is approximately (1 + 1/n)ⁿ as n → infinity.