How trigonometric values are calculated
On the unit circle, an angle θ measured from the positive x-axis lands on a point whose coordinates are (cos θ, sin θ). The other four functions are built from ratios and reciprocals of sin θ and cos θ.
- θ — input angle, in degrees or radians
- sin θ, cos θ — y, x coordinates of the point on the unit circle
- tan θ — sin θ ÷ cos θ — undefined where cos θ = 0
- csc θ, sec θ, cot θ — 1/sin θ, 1/cos θ, cos θ/sin θ
More detail
Why some values show "—"
tan θ and sec θ divide by cos θ, so they're undefined whenever cos θ = 0 — at 90°, 270°, and every 90° + 180°n after that. csc θ and cot θ divide by sin θ, so they're undefined at 0°, 180°, and every 180°n. This calculator shows "—" for those rather than a misleading number.
Degrees vs. radians. Switch the Unit field to change how the angle above is read — the calculator doesn't convert your number, it reinterprets it. 30 degrees and π/6 radians are the same angle, but the rounded 0.5236 is really 30.0001° — close enough that five outputs match while cot θ reads 1.7320 instead of 1.7321. Type 0.5235988 or more digits to match all six. The line under sin θ shows the equivalent value in the other unit.
Frequently asked questions
What are sin, cos, and tan of 30 degrees?
sin 30° = 0.5, cos 30° ≈ 0.8660, and tan 30° ≈ 0.5774 — these are the calculator's defaults, so entering 30 with Unit set to Degrees reproduces them directly.
Why is tan 90° undefined?
tan θ = sin θ / cos θ, and cos 90° = 0. Dividing by zero has no defined result, so the calculator shows "—" for tan and sec at 90° (and every 90° + 180°n).
How do I calculate with radians instead of degrees?
Set the Unit field to Radians and enter the angle in radians directly — for example, π/6 typed as 0.5235988 gives the same six values as 30 degrees. Rounding it to 0.5236 is 30.0001°, already enough to move cot θ to 1.7320, so add digits when every value has to match. The line below sin θ shows the same angle converted to the other unit.
How are csc, sec, and cot related to sin, cos, and tan?
csc θ = 1/sin θ and sec θ = 1/cos θ. cot θ = cos θ/sin θ, which equals 1/tan θ wherever tan θ is defined — at 90°, where tan θ is undefined, cot θ is still 0. For 30°, sin = 0.5 so csc = 2; cos ≈ 0.8660 so sec ≈ 1.1547; tan ≈ 0.5774 so cot ≈ 1.7321.