Mistakes · Trigonometry
5 Common Trig Mistakes and How to Catch Them
Four of the five are caught by one habit: check the sign against the quadrant, and the size against the reference angle.
1. The wrong calculator mode
sin(30) is 0.5 in degree mode and −0.988 in radian mode. Both are correct answers to different questions, which is what makes this error so easy to miss: the answer looks like a perfectly ordinary number.
| Input | Degree mode | Radian mode |
|---|---|---|
| sin 30 | 0.5 | −0.988 |
| cos 60 | 0.5 | −0.952 |
| tan 45 | 1 | 1.620 |
The check: for an angle you know, like sin 30°, the answer should be 1/2. If a calculator disagrees, the mode is wrong.
2. Losing the sign from the quadrant
cos 135° is −√2/2, not +√2/2. The reference angle 45° gives the size, and the quadrant gives the sign. Copying the value from the first quadrant without the sign is the single most common error.
3. Using the wrong reference angle
The reference angle is not always the angle minus something the same way. It depends on the quadrant.
| Quadrant | Angle θ | Reference angle |
|---|---|---|
| I | 0° to 90° | θ |
| II | 90° to 180° | 180° − θ |
| III | 180° to 270° | θ − 180° |
| IV | 270° to 360° | 360° − θ |
For 150°, the reference angle is 180° − 150° = 30°, not 150° − 90° = 60°. Using the wrong one gives the right-looking value for the wrong angle.
4. Treating an undefined value as zero
tan 90° is undefined, not 0. Tangent is sine divided by cosine, and cosine is 0 at 90°, so the division has no answer. It is also not infinity as a number you can use in further arithmetic: the correct statement is that the value does not exist.
The same goes for secant at 90° and 270°, and for cosecant and cotangent at 0° and 180°.
5. Not reducing a large or negative angle first
sin 750° looks hard, but 750° − 720° = 30°, so it equals sin 30° = 1/2. A negative angle like −30° becomes 330° after adding 360°, and sin 330° = −1/2. Skipping the reduction leaves an angle you cannot find a quadrant for.
Why a quick check catches most of these
Four of the five mistakes show up as a mismatch between the answer and the quadrant or the reference angle, so a two-second check is enough. Ask whether the sign agrees with the quadrant, and whether the size matches the reference angle's value from the standard table. To check your answer when an angle is not standard, use a second method such as the identity sin² + cos² = 1: if it does not hold, a value is wrong.
The Trig Calculator on this site gives the quadrant and the reference angle at every step, so each of these slips becomes visible instead of hidden inside a single number.
Three smaller slips worth naming
The first is mixing degrees and radians inside one problem. If an angle is given as π/6, keep working in radians, or convert it to 30° once and stay in degrees. Switching halfway is how a correct reference angle ends up paired with the wrong table.
The second is forgetting that a coterminal angle gives the same value. 30°, 390° and −330° all land on the same terminal side, so they share every trig value. The check is to reduce to between 0° and 360° first, then compare.
The third is forgetting that cosecant, secant and cotangent are reciprocals. csc 210° is 1 ÷ sin 210° = 1 ÷ (−1/2) = −2. When an exact value for the original function exists, the reciprocal's exact value follows by flipping it, and the sign carries over unchanged. Keeping the exact value as a fraction, rather than a rounded decimal, makes that flip easy.
Questions about common trig mistakes
Why is my calculator giving the wrong value for sin 30?
Most likely it is in radian mode. sin 30 is 0.5 in degree mode and −0.988 in radian mode.
Is tan 90° equal to zero?
No. It is undefined, because tan = sin ÷ cos and cos 90° = 0, which makes the division impossible.
How do I know the sign of a trig value?
From the quadrant: sine is positive in I and II, cosine in I and IV, tangent in I and III.
How do I handle an angle bigger than 360°?
Subtract 360° until it is between 0° and 360°. sin 750° = sin 30° = 1/2.