How to Evaluate Trigonometric Functions in Degree Mode

Begin with the angle unit

A calculator interprets the same numerical input differently in degree and radian modes. If a problem writes an angle with a degree symbol, select degree mode before evaluating the function. If the input contains $\pi$ or explicitly says radians, use radian mode.

When software accepts only radians, convert degrees with

$$\theta_{\text{rad}}=\theta_{\text{deg}}\frac{\pi}{180}.$$

Use a repeatable calculator workflow

1. Confirm degree or radian mode.
2. Enter the trigonometric function and the complete angle.
3. Keep several displayed digits during any intermediate work.
4. Round once, at the end, to the place requested.

For example, in degree mode,

$$\sin 22^\circ\approx0.3746066,$$

which rounds to $0.375$ to the nearest thousandth. The unrounded display, not an earlier shortened value, should be used in any later calculation.

Check the sign and range

The unit circle gives quick error checks:

• sine and cosine always lie between $-1$ and $1$;
• first-quadrant sine, cosine, and tangent values are positive;
• second-quadrant sine is positive, while cosine and tangent are negative.

For instance, $125^\circ$ lies in the second quadrant, so $\tan 125^\circ$ must be negative. A positive calculator result would signal a mode or entry mistake.

Know what rounding changes

Rounding to the nearest hundredth means examining the thousandths digit. A value of $0.7849\ldots$ becomes $0.78$, while $0.7851\ldots$ becomes $0.79$. Keep guard digits until the requested final precision is known, especially when a trigonometric value feeds a later side-length or angle calculation.