Why is tan 90° undefined?

Tangent is sine divided by cosine, and cos 90° is 0, so the ratio has no finite value — the line through the origin becomes vertical on the unit circle.

Which angles are worth memorising?

0°, 30°, 45°, 60° and 90°. Their sines follow √0/2, √1/2, √2/2, √3/2 and √4/2, so 30° gives 0.5 and 60° gives 0.8660254.

Do the values change with the size of the triangle?

No. Sine, cosine and tangent are ratios, so a 3-4-5 and a 6-8-10 triangle share them; only the side lengths scale.

📐 Trigonometric Calculator (Degree)

Calculate trigonometric function values based on degrees in an equilateral triangle.

Unit Circle: sin = height, cos = base

Draw a right triangle inside a unit circle (radius = 1). The horizontal base always equals cos θ and the vertical height always equals sin θ — because the hypotenuse IS the radius (= 1).

θ = 45°

Drag the slider to change the angle

cos θ = 0.7071base = cos θ

sin θ = 0.7071height = sin θ

r = 1hypotenuse = radius = 1

Angle (°)

What are Trigonometric Functions?

Ratios of sides in a unit right triangle (hypotenuse=1) for a given angle. With hypotenuse set to 1:

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent = sin θ / cos θ

Reciprocal Functions

csc θ = 1 / sin θ (cosecant)

sec θ = 1 / cos θ (secant)

cot θ = 1 / tan θ (cotangent)

Special Angle Values

θ	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	∞

Pythagorean Identity

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Worked examples

At 30° the sine is 0.5, the cosine 0.8660254 and the tangent 0.5773503, while the cosecant is 2 and the secant about 1.1547005.

At 45° sine and cosine are both 0.7071068 and the tangent is exactly 1, because the reference triangle has two equal legs.

What the values assume

Tangent is undefined at 90° and every 180° after it, so the calculator marks those angles instead of showing a huge number. Degrees are converted to radians internally, which can show up in the last digits. The values describe a right triangle with hypotenuse 1, which is also the height and horizontal offset of the point on the unit circle.

Frequently asked questions

Why is tan 90° undefined?: Tangent is sine divided by cosine, and cos 90° is 0, so the ratio has no finite value — the line through the origin becomes vertical on the unit circle.
Which angles are worth memorising?: 0°, 30°, 45°, 60° and 90°. Their sines follow √0/2, √1/2, √2/2, √3/2 and √4/2, so 30° gives 0.5 and 60° gives 0.8660254.
Do the values change with the size of the triangle?: No. Sine, cosine and tangent are ratios, so a 3-4-5 and a 6-8-10 triangle share them; only the side lengths scale.