CALCULATE WITH CONTEXT
Scientific calculations, with a clear method
Evaluate trigonometry, logarithms, powers, roots and counting expressions. Check the angle mode, enter a complete expression, then use the worked examples to make sense of the result.
Use function names, brackets and commas, or the keypad below.
Keyboard inside calculator: numbers, + − * / ^ ( ) , ! %, Enter to evaluate, Backspace to delete, Escape to clear. You can type complete function names in the Expression field.
How to use the scientific calculator
- Choose DEG or RAD first. Match the angle unit in your question. Use DEG for sin(30°), or RAD for sin(π/6). The choice also controls the unit returned by inverse trigonometric functions.
- Type an expression or use the buttons. For example, enter
sqrt(7^2+24^2). Function keys insert an opening bracket; close it with ). Use the comma key to separate the two inputs innCr(8,3)ornroot(3,-27). - Evaluate, then check the meaning. Press Calculate, = or Enter. Confirm the sign, approximate size and units before copying your answer. A numerical answer alone does not choose the right formula for a word problem.
- Keep precision until the final step. ANS recalls the last completed result. M+ adds the current valid value to memory, M− subtracts it, MR inserts the stored value, and MC clears memory. C clears the current expression. History keeps the last 20 completed calculations in this page session; choose an entry to recall its result.
The expression field accepts function names such as sin, asin, log, ln and sqrt. Use PI for π and EULER for e. For a fraction, type division, such as (2/3)+(1/4). The result is a decimal, not a symbolic fraction. For fraction-specific calculations, use the fractions calculator.
6/(2*(1+2)) if the entire product belongs in the denominator. Avoid ambiguous expressions such as 6÷2(1+2). This calculator gives implicit multiplication the same priority as explicit multiplication and division.What the scientific keys mean
Angles, trigonometry and inverse functions
In a right triangle, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. For angles beyond a right triangle, the unit circle extends these definitions. One full turn is 360° or 2π radians, so 180° = π radians.
sin⁻¹ is an inverse function, not a reciprocal. Enter asin(0.5) to get 30 in DEG mode or approximately 0.523599 in RAD mode. In contrast, 1/sin(30) in DEG mode equals 2. Inverse sine and cosine require inputs between −1 and 1. Their principal outputs lie in [−90°, 90°] and [0°, 180°], respectively; inverse tangent lies between −90° and 90°. The corresponding radian intervals use π/2 and π.
Logarithms and exponential functions
log means base 10: log(1000) = 3 because 10³ = 1000. ln means base e: ln(e) = 1. Both require a positive input for a real result. The eˣ and 10ˣ keys reverse the corresponding logarithms. To calculate a different base, use a ratio: log₂(32) = ln(32) ÷ ln(2) = 5.
Powers, roots and signs
The xʸ key inserts ^. Powers are grouped from the right: 2^3^2 means 2^(3²) = 512. Exponentiation is performed before an unbracketed leading minus: -2^2 = −4, but (-2)^2 = 4. A negative exponent takes a reciprocal: 2⁻³ = 1/8.
The square-root key returns the nonnegative square root. Thus √9 = 3, while the equation x² = 9 has two solutions, x = ±3. The cube root of −27 is −3. Use nroot(n,x) for an integer-index root; an even root of a negative number has no real result. Negative indices mean reciprocals: nroot(−2,16) = 1/4. Zero is not an allowed root index.
Factorials, combinations and permutations
For a nonnegative integer n, n! counts orderings of n distinct items: 5! = 5×4×3×2×1 = 120, and 0! = 1. This tool's factorial key accepts integers from 0 to 170. A value such as 2.5! is outside that key's domain and produces an error; it is not rounded to a nearby integer.
For choosing r distinct objects from n without replacement, nCr = n! ÷ [r!(n−r)!] counts selections where order does not matter. nPr = n! ÷ (n−r)! counts ordered selections. Both need whole numbers with 0 ≤ r ≤ n. Inputs must not exceed 9,007,199,254,740,991, and a result outside the calculator’s finite range is rejected. Repetition requires a different model; for example, a four-digit code allowing repeated digits has 10⁴ possibilities, not 10P4.
EXP, percentages and numerical limits
EXP enters a power-of-ten exponent. Type 3.2E6 or press 3.2, EXP, 6 for 3.2×10⁶. This is one number. It is different from the e key, which inserts the constant e ≈ 2.71828. For very small values, use a negative exponent: 4.5E−3 = 0.0045. See the scientific notation examples for converting and combining such numbers.
The percent key divides its preceding value by 100. Therefore 15% = 0.15 and 200×15% = 30. It does not use the contextual percentage behavior of some pocket calculators: 200+15% is 200.15. For a 15% increase, use 200*(1+15%) = 230, or the percentage calculator.
Calculations use finite-precision real numbers and display up to 12 significant digits. Large integer counts may be approximations; displayed rounding is not a guarantee of 12 accurate digits after every operation. Overflow, undefined real operations and malformed expressions produce errors. Very small results may underflow to zero. The calculator does not provide complex arithmetic, symbolic algebra, unit conversion or certified engineering results.
Seven worked examples
1. Find a ladder's vertical reach
A 4 m ladder makes a 65° angle with level ground. The vertical height is opposite the angle, and the ladder is the hypotenuse. Therefore h/4 = sin 65°, so h = 4 sin 65°.
DEG: 4*sin(65) ≈ 3.625231148 m ≈ 3.63 m
The answer is less than 4 m, as it must be. Keep the unrounded sine until the final multiplication.
2. Recover an angle from a ratio
A ramp rises 3 m over a horizontal distance of 8 m. Opposite ÷ adjacent = 3/8, so θ = tan⁻¹(3/8).
DEG: atan(3/8) ≈ 20.56°
This uses inverse tangent because the known sides are the rise and horizontal run. It would be wrong to use the ramp's sloping length as the adjacent side.
3. Solve a doubling equation
Suppose an idealized quantity is multiplied by 1.06 each year. To find a doubling time, solve 1.06ᵗ = 2. Taking logs gives t log(1.06) = log(2).
log(2)/log(1.06) ≈ 11.89566105 years
The exponential model permits fractional years. If the quantity changes only at whole-year updates, it first reaches twice its original size at year 12: 1.06¹¹ ≈ 1.8983 and 1.06¹² ≈ 2.0122. For a specific repayment calculation, use the loan calculator and its stated assumptions.
4. Count unordered and ordered selections
A class chooses 3 representatives from 8 students. Each group is counted once regardless of order.
nCr(8,3) = (8×7×6)/(3×2×1) = 56
If the same students instead fill three distinct roles, order matters: nPr(8,3) = 8×7×6 = 336. Similarly, choosing 6 distinct numbers from 49 gives nCr(49,6) = 13,983,816 unordered sets. This is a count, not a probability until the sample space and event are defined. Continue with the probability worksheets.
5. Find a rectangle's diagonal
A rectangle has side lengths 7 cm and 24 cm. Its diagonal forms a right triangle, so d² = 7² + 24² = 625.
sqrt(7^2+24^2) = √625 = 25 cm
The brackets keep the whole sum under the square root. A length is nonnegative, so the required diagonal is 25 cm.
6. Divide numbers in scientific notation
Using an approximate wave speed of 3.0×10⁸ m/s and frequency 1.5×10⁸ Hz, wavelength λ = speed ÷ frequency. Since 1 Hz = 1/s, the resulting unit is metres.
3E8/1.5E8 = (3/1.5)×10⁰ = 2.0 m
The two powers of ten cancel. The speed used here is rounded; do not treat extra calculator digits as extra measurement accuracy.
7. Evaluate a signed power and an odd root
First evaluate the power: −2³ = −8. Then ∛(−27) = −3 because (−3)³ = −27.
-2^3+cbrt(-27) = −8 + (−3) = −11
For equations involving unknown values rather than a numerical expression, try the algebra calculator.
Practice, then reveal the reasoning
Write an estimate before calculating. Use DEG unless the question specifies RAD.
1. Evaluate 6 + 4 × 3
18. Multiply first: 4×3 = 12, then 6+12 = 18. The brackets in (6+4)×3 would instead give 30.
2. Evaluate −3² and (−3)²
−9 and 9. In −3² the minus is outside the square; in (−3)² it is part of the base.
3. Find cos(60°)
0.5. Select DEG and enter cos(60). In RAD, the equivalent entry is cos(PI/3).
4. In RAD, find sin(π/2)
1. Enter sin(PI/2) in RAD. π/2 radians is a quarter turn, or 90°.
5. Find the principal angle with sine 0.5
30°, or π/6 radians. Use asin(0.5). The principal result is not every solution; 150° also has sine 0.5.
6. Evaluate log(0.001)
−3. The positive input is 10⁻³. A negative logarithm is valid; a nonpositive input is outside the real logarithm's domain.
7. Evaluate 0! + 5!
121. 0! = 1 and 5! = 120, so their sum is 121.
8. Choose 2 students from 10 with no roles
45 groups. nCr(10,2) = 10×9÷2. Dividing by 2 removes the two orders of each pair.
9. Give two distinct roles to students from the same group of 10
90 assignments. There are 10 choices for the first role and 9 for the second: nPr(10,2) = 90.
10. Evaluate 6E−3 ÷ 2E−2
0.3. (6÷2)×10⁻¹ = 3×0.1. Use the exponent sign inside each scientific-notation number.
11. Find 12.5% of 240
30. 12.5% = 0.125 = 1/8, and 240÷8 = 30. Enter 240*12.5%.
12. Which have real outputs: √(−16), ∛(−8), log(0), tan(90°)?
Only ∛(−8), which equals −2. A real square cannot be negative; real logarithms need positive inputs; tangent is undefined at 90° because cosine is zero.
Common mistakes and questions
Why does my answer differ from another calculator?
Check angle mode, bracket placement, scientific-notation entry and percent behavior first. Compare the full expression rather than just the last result. If you rounded an intermediate number, ANS may produce a more accurate final value than retyping the displayed digits.
Why does the calculator reject 2.5!?
This factorial button implements the whole-number counting definition. Extensions such as the gamma function are a different operation and are not provided here. Changing 2.5 to 3 would change the question.
Why do some trig results show tiny decimals near zero?
Binary floating-point arithmetic cannot represent every decimal or π exactly. This calculator explicitly handles recognized cardinal angles, but other equivalent expressions can accumulate small errors. Rounding to a fixed number of significant digits alone does not turn every tiny value into zero; a genuinely small result must stay small.
Can I use it to solve equations or display fractions?
It evaluates numerical expressions to real decimal approximations. It does not rearrange equations, prove identities or display exact surds and rational fractions. For example, sqrt(2) is a decimal approximation even though √2 is an exact mathematical value.
What do C, ANS, memory and History retain?
C clears the expression while ANS retains the most recent completed answer. MC clears the separate memory value. History and memory belong to the current page session and reset when the page reloads. Do not use them as permanent storage.
Does RAN generate a suitable value for security or research?
RAN inserts a pseudorandom decimal from 0 up to, but not including, 1. It is convenient for informal examples, not cryptographic secrets or reproducible statistical sampling.
If your problem involves hours, minutes or elapsed time, the time calculator handles those units directly.
Sources & References
- OpenStax Precalculus 2e: Angles: Degree and radian measures.
- OpenStax: Unit Circle, Sine and Cosine: Trigonometric definitions and values.
- OpenStax: Inverse Trigonometric Functions: Principal angles and real domains.
- OpenStax: Logarithmic Functions: Logarithms, exponentials and positive arguments.
- OpenStax: Logarithmic Properties: Change of base and logarithmic rules.
- OpenStax: Counting Principles: Factorials, combinations and permutations.
- ECMAScript specification: Number type: Binary64 arithmetic and its numerical limits.
- BIPM: SI defining constants: Exact vacuum speed of light and SI context.
Original examples, illustrations and practice explanations by HeLovesMath. References checked October 4, 2026.
