DEG vs RAD (angle mode) — why it matters
Trigonometry functions depend on your angle unit. The same input number can mean degrees or radians — and you’ll get completely different results.
Scientific calculators usually have an angle mode toggle: DEG (degrees) or RAD (radians). Trig functions like sin, cos, and tan interpret inputs in the current mode.
A common mistake is typing sin(90) expecting 1, while the calculator is in RAD mode. In radians, 90 is a huge angle (about 5157°), so sin(90) is not 1.
Rule of thumb: use DEG for geometry, everyday angle intuition (30°, 45°, 90°). Use RAD for calculus/physics formulas and when problems explicitly say “radians”.
- In DEG: sin(30) = 0.5, sin(90) = 1
- In RAD: sin(π/2) = 1 (≈ sin(1.5708))
Sanity check: π radians = 180°
Before trig, confirm whether you’re in DEG or RAD mode.
Powers & roots (x², xʸ, ⁿ√x)
Powers grow (or shrink) values quickly. Roots are the inverse of powers — but some combinations are not real numbers.
Powers (exponents) show repeated multiplication. For example, x² means x × x. A general power xʸ means multiply x by itself y times (for integer y).
Roots undo powers: √x is the number that squared gives x. More generally, ⁿ√x is the number that raised to the power n equals x.
Not every input has a real-number answer: √(−1) is not a real number (it’s imaginary). Many scientific calculators will show a domain error for that case when complex numbers are disabled.
- 2^5 = 32
- ³√27 = 3
- √81 = 9
Domain note: even root of a negative is not real
Powers and roots are inverses, but real-number domain rules still apply.
Logs & exponentials (log, ln, 10ˣ, eˣ)
Logarithms undo exponentials. Most calculators use log = log base 10, and ln = log base e.
The function log(x) commonly means log base 10. The function ln(x) means natural log (base e ≈ 2.71828).
Exponentials are the inverse operations: 10ˣ undoes log, and eˣ undoes ln. This shows up everywhere in growth/decay problems and continuous compounding.
Domain rule: logs require positive inputs. log(0) and log(−5) are undefined in real numbers, so calculators usually return an error.
- log(100) = 2
- ln(e) = 1
- e^0 = 1
log vs ln
Logs and exponentials are inverse pairs: log ↔ 10ˣ, ln ↔ eˣ.
Order of operations + parentheses
Multiplication/division happen before addition/subtraction — unless parentheses change the grouping.
Most scientific calculators follow standard precedence rules: parentheses first, then powers, then multiplication/division, then addition/subtraction.
This means 2 + 3 × 4 evaluates as 2 + (3 × 4) = 14. But (2 + 3) × 4 evaluates as 20. If results look “wrong”, check your parentheses.
When using a full expression input, it’s usually better to add parentheses explicitly for clarity — especially in longer formulas.
- 2 + 3 × 4 = 14
- (2 + 3) × 4 = 20
Add parentheses when you care about grouping
Parentheses override precedence. When in doubt, add them.
Quick Reference
| Function | Meaning | Example |
|---|---|---|
| sin(x) | Sine of angle x | sin(30°) = 0.5 |
| cos(x) | Cosine of angle x | cos(60°) = 0.5 |
| tan(x) | Tangent of angle x | tan(45°) = 1 |
| log(x) | Base-10 logarithm | log(100) = 2 |
| ln(x) | Natural log (base e) | ln(e) = 1 |
| xʸ | Power/exponent | 2^5 = 32 |
| ⁿ√x | n-th root | ³√27 = 3 |
| π | Pi constant | π ≈ 3.14159 |
| e | Euler’s number | e ≈ 2.71828 |
FAQs
What’s the difference between DEG and RAD?
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What’s the difference between DEG and RAD?
▾They are two different angle units. In DEG, 180° is a straight line. In RAD, π is a straight line. Trig functions use whichever unit your calculator is set to.
Why is sin(90) not equal to 1 on my calculator?
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Why is sin(90) not equal to 1 on my calculator?
▾Your calculator is probably in RAD mode. In radians, the input 90 is not 90°. Switch to DEG, or use sin(π/2) in RAD.
Is “log” the same as “ln”?
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Is “log” the same as “ln”?
▾Usually no. On most scientific calculators, log means base 10 and ln means base e. Always check the label in your calculator UI.
Why does √(−9) show an error?
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Why does √(−9) show an error?
▾Because the square root of a negative number is not a real number. If complex numbers are not supported, the calculator should treat it as an invalid input.
Why do parentheses change the answer?
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Why do parentheses change the answer?
▾Parentheses change grouping. Without parentheses, multiplication/division happen before addition/subtraction (standard precedence rules).
What does “domain error” mean?
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What does “domain error” mean?
▾It means the function is not defined for that input in real numbers (for example, log(0), log(−5), or division by 0).