Question 1

What is sin(90°)?

Accepted Answer

sin(90°) = 1.

This is the maximum value sine can ever reach. On the unit circle, 90° is the point (0, 1) — the very top of the circle. The y-coordinate is sin(90°) = 1.

sin(90°) = 1 makes intuitive sense from the right triangle definition. As the angle approaches 90°, the opposite side approaches the hypotenuse in length, making the ratio opposite/hypotenuse approach 1.

At exactly 90°, the 'triangle' has collapsed — the opposite side equals the hypotenuse and the adjacent side has length 0. Sin = 1/1 = 1.

Question 2

What is cos(90°)?

Accepted Answer

cos(90°) = 0.

On the unit circle, 90° is the point (0, 1). The x-coordinate is cos(90°) = 0.

From the right triangle: as the angle approaches 90°, the adjacent side shrinks to zero while the hypotenuse stays fixed. So cos = adjacent/hypotenuse → 0/hypotenuse = 0.

cos(90°) = 0 is the zero crossing of cosine. Cosine starts at 1 (at 0°), decreases through 0 (at 90°), reaches -1 (at 180°), returns through 0 (at 270°), and comes back to 1 (at 360°).

Note: cos(90°) = sin(0°) = 0 by the co-function identity.

Question 3

Why is tan(90°) undefined?

Accepted Answer

tan(90°) = sin(90°)/cos(90°) = 1/0, which is undefined.

Division by zero is undefined in mathematics — there is no finite number that equals 1/0.

Geometric interpretation: tangent represents the slope of the radius line on the unit circle. At 90°, the radius is pointing straight up — a vertical line. A vertical line has no finite slope (it is 'infinitely steep'), which is why tan(90°) is undefined.

As the angle approaches 90° from below (89°, 89.9°, 89.99°...):
tan → +∞

As the angle approaches 90° from above (91°, 90.1°, 90.01°...):
tan → -∞

This jump from +∞ to -∞ is the vertical asymptote of the tangent function at 90°. Tan has a vertical asymptote at every odd multiple of 90° (90°, 270°, 450°...).

Question 4

What is cot(90°) and why is it 0?

Accepted Answer

cot(90°) = cos(90°)/sin(90°) = 0/1 = 0.

Unlike tan(90°) which is undefined, cot(90°) is perfectly well-defined and equals 0.

This is because cot = cos/sin, and at 90°, cos = 0 while sin = 1. Zero divided by a non-zero number is 0, not undefined.

Cotangent is the reciprocal of tangent. As tan(90°) approaches infinity, cot approaches zero — they are reciprocals. But at the exact boundary:
- tan(90°) is undefined (denominator is 0)
- cot(90°) = 0 (numerator is 0, denominator is 1)

Pattern: cot(0°) is undefined (cos/sin = 1/0), but cot(90°) = 0. This is the reverse of tan: tan(0°) = 0, tan(90°) = undefined.

Question 5

What happens to trig functions just past 90°?

Accepted Answer

At 90° the angle crosses from Quadrant I into Quadrant II. The sign rules change:

Quadrant I (0° to 90°): sin+, cos+, tan+
Quadrant II (90° to 180°): sin+, cos-, tan-

Exact values at key nearby angles:
- sin(120°) = sin(60°) = √3/2 (sin stays positive)
- cos(120°) = -cos(60°) = -1/2 (cos turns negative)
- tan(120°) = -tan(60°) = -√3 (tan turns negative)

For any angle θ in Quadrant II: sin(θ) = sin(180°-θ) and cos(θ) = -cos(180°-θ).

So 91° has: sin(91°) = sin(89°) ≈ 0.9998, cos(91°) = -cos(89°) ≈ -0.0175. The sign of cosine flips immediately past 90°.

Function	Exact Value	Decimal
sin(90°)	1	1
cos(90°)	0	0
tan(90°)	Undefined	—
csc(90°)	1	1
sec(90°)	Undefined	—
cot(90°)	0	0

NAME	FORMULA	RESULT
Pythagorean identity at 90°	sin²(90°)+cos²(90°) = 1²+0² = 1+0	= 1 ✓
Tan(90°) undefined	tan(90°) = sin(90°)/cos(90°) = 1/0	= Undefined (division by zero)
Csc(90°)	csc(90°) = 1/sin(90°) = 1/1	= 1
Sec(90°) undefined	sec(90°) = 1/cos(90°) = 1/0	= Undefined (division by zero)
Cot(90°)	cot(90°) = cos(90°)/sin(90°) = 0/1	= 0
Co-function identity	sin(90°) = cos(0°) = 1	90° and 0° are complementary
Maximum of sine	sin(θ) reaches maximum of 1 at θ = 90°	sin(90°) = 1 is the global maximum
Double angle (sin)	sin(180°) = 2·sin(90°)·cos(90°) = 2·1·0	= 0 ✓

Sin Cos Tan of 90 Degrees — Exact Values

TRIG VALUES AT 90°

UNIT CIRCLE

FORMULAS & IDENTITIES

WORKED EXAMPLE

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

MORE TRIGONOMETRY CALCULATORS

°	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	Undef
120°	√3/2	−1/2	−√3
135°	√2/2	−√2/2	−1
150°	1/2	−√3/2	−1/√3
180°	0	−1	0
270°	−1	0	Undef
360°	0	1	0