Degrees vs Radians: Why Your Calculator Gives the Wrong sin Answer

The short answer

If sin(30) gives −0.9880316241 instead of 0.5, your calculator is in radian mode and is treating 30 as 30 radians, not 30 degrees. Switch to degree mode and the answer becomes 0.5. In the Scientific Calculator, the top-left button shows the current mode (DEG or RAD); tap it to switch. The calculator starts in DEG mode.

Use degrees for everyday geometry, school trigonometry, navigation and building angles. Use radians for calculus, physics formulas and most programming languages.

What degrees and radians are

Both measure the size of an angle.

• Degrees split a full turn into 360 equal parts. A right angle is 90°.
• Radians measure an angle by arc length on a circle of radius 1. A full turn is 2π radians, about 6.2832.

Angle	Degrees	Radians
Full turn	360°	2π ≈ 6.2832
Half turn	180°	π ≈ 3.1416
Right angle	90°	π/2 ≈ 1.5708
Two-thirds of a right angle	60°	π/3 ≈ 1.0472
Half a right angle	45°	π/4 ≈ 0.7854
One-third of a right angle	30°	π/6 ≈ 0.5236
One radian	≈ 57.2958°	1

The two conversions you need:

• Degrees to radians: multiply by π/180. Example: 45 × π/180 ≈ 0.7854.
• Radians to degrees: multiply by 180/π. Example: 1 × 180/π ≈ 57.2958°.

The same button, two very different answers

Because one radian is about 57°, typing a degree value in radian mode gives an answer for a completely different angle. These are the results from the Scientific Calculator:

You type	In DEG mode	In RAD mode
sin(30)	0.5	−0.9880316241
cos(60)	0.5	−0.9524129804
tan(45)	1	1.619775191
sin⁻¹(0.5)	30	0.5235987756

A quick way to check your mode: work out sin(30). If you get 0.5, you are in degrees. If you get −0.988, you are in radians.

Notice the last row. Inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) return an angle, so the mode controls the unit of the answer. In DEG mode, sin⁻¹(0.5) gives 30 (degrees); in RAD mode it gives 0.5236 (radians), which is the same angle written differently.

How to use the Scientific Calculator's trig functions

Check the mode label at the bottom-left of the display, or on the top-left button. Tap the button to switch between DEG and RAD.

Tap sin, cos or tan. The calculator writes the function with an opening bracket, for example sin(.

Type the angle, then tap ) to close the bracket.

Tap =.

If you forget the closing bracket, the calculator shows Error. Tap ⌫ to delete the last character, or AC to clear everything, and try again.

Enter a single number inside the function. If the angle is a sum, such as 30 + 15, work it out first and enter sin(45). Typing an expression inside the bracket in DEG mode does not always give the right answer, so calculating the angle first is the safer habit.

Use π in RAD mode only. sin(π/6) is 0.5 in RAD mode, because π/6 is already in radians. In DEG mode, the calculator would treat π/6 (about 0.5236) as degrees and give a very small number.

Why cos(90) does not show exactly 0

In DEG mode, cos(90) on the Scientific Calculator shows 6.123233996e-17, not 0. Similarly, sin(180) shows 1.224646799e-16, and tan(90) shows a huge number, 16331239350000000, instead of an error.

These are not real mistakes in the maths. The calculator converts 90° to radians by multiplying by π, and computers can only store π to about 16 significant digits. The angle it actually works with is a tiny fraction away from exactly 90°, so the cosine is a tiny fraction away from 0.

Read the results like this:

• 6.123233996e-17 means 6.12 × 10⁻¹⁷, or 0.0000000000000000612. Treat it as 0.
• 1.224646799e-16 for sin(180) is also 0 for any practical purpose.
• tan(90) is undefined. The huge number is the calculator dividing by an almost-zero cosine. Treat it as "no answer".

Many handheld calculators hide this by rounding or by recognising exact angles. General-purpose calculators, spreadsheets and programming languages often show the raw floating-point result, so it helps to recognise it.

Where each unit is the default

Tool	Default angle unit
Lazyblink Scientific Calculator	Degrees (switchable)
JavaScript Math.sin	Radians
Python math.sin	Radians
Excel and Google Sheets SIN	Radians

In spreadsheets and code, convert first: in Excel or Google Sheets, =SIN(RADIANS(30)) returns 0.5. In Python, math.sin(math.radians(30)) does the same. Passing degrees straight into these functions is the same mistake as using a calculator in the wrong mode.

When to use degrees and when to use radians

Use degrees for:

• School geometry and trigonometry problems that give angles in degrees
• Slopes, roof pitches, ramps and other construction angles
• Compass bearings and map directions
• Angles measured with a protractor

Use radians for:

• Calculus, where formulas such as the derivative of sin x being cos x only hold in radians
• Physics formulas for rotation, waves and oscillation (angular speed is in radians per second)
• Programming and spreadsheet trig functions
• Any problem where angles are written with π, such as π/4

If a question has π in the angle, it is almost certainly in radians. If it has a ° symbol, use degrees.

Common mistakes

• Leaving the calculator in RAD mode after a physics or calculus problem and then doing a geometry question.
• Mixing units in one problem, for example converting part of an answer to radians and forgetting the rest.
• Reading 6.1e-17 as 6.1. The e-17 part makes it essentially zero.
• Typing degree values with π. 30° is π/6 radians, not 30π.

Related tools

Convert any angle between degrees, radians and other units with the Angle Converter. For powers, logs and factorials, read how to use a scientific calculator, and solve equations with the Quadratic Equation Solver.