Question 1

What is sin(30°)?

Accepted Answer

sin(30°) = 1/2 = 0.5.

This is one of the most important values in trigonometry. It comes from the 30-60-90 triangle, which has sides in the ratio 1 : √3 : 2.

sin = opposite / hypotenuse = 1/2

On the unit circle, 30° corresponds to the point (√3/2, 1/2). The y-coordinate is sin(30°) = 1/2.

Memory trick: for angles 0°, 30°, 45°, 60°, 90°, the sine values are sqrt(0)/2, sqrt(1)/2, sqrt(2)/2, sqrt(3)/2, sqrt(4)/2 — which simplify to 0, 1/2, √2/2, √3/2, 1.

Question 2

What is cos(30°)?

Accepted Answer

cos(30°) = √3/2 ≈ 0.86602540.

From the 30-60-90 triangle with sides 1, √3, 2:
cos = adjacent / hypotenuse = √3/2

On the unit circle, 30° is the point (√3/2, 1/2), so the x-coordinate is cos(30°) = √3/2.

Note: cos(30°) = sin(60°) — the cosine of any angle equals the sine of its complement. Since 30° + 60° = 90°, cos(30°) = sin(60°) = √3/2.

Question 3

What is tan(30°)?

Accepted Answer

tan(30°) = 1/√3 = √3/3 ≈ 0.57735027.

tan = sin/cos = (1/2) / (√3/2) = 1/√3

Rationalising: 1/√3 × √3/√3 = √3/3

Alternatively: in a 30-60-90 triangle, tan = opposite/adjacent = 1/√3 (short leg over long leg).

Compare: tan(45°) = 1, tan(60°) = √3. The pattern for tan at 30°, 45°, 60° is: 1/√3, 1, √3 — each is the reciprocal of the next in the sequence.

Question 4

What are the exact values for all 6 trig functions at 30°?

Accepted Answer

All six trigonometric functions at 30 degrees:

Primary functions:
- sin(30°) = 1/2 = 0.5
- cos(30°) = √3/2 ≈ 0.866025
- tan(30°) = 1/√3 = √3/3 ≈ 0.577350

Reciprocal functions:
- csc(30°) = 1/sin = 2
- sec(30°) = 1/cos = 2/√3 = 2√3/3 ≈ 1.154701
- cot(30°) = 1/tan = √3 ≈ 1.732051

Verification: sin²(30°) + cos²(30°) = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1 ✓

Note: cot(30°) = √3 = tan(60°), and csc(30°) = 2 = sec(60°). The functions at 30° and 60° are swapped reciprocals — a direct consequence of them being complementary angles.

Question 5

Why is sin(30°) = cos(60°)?

Accepted Answer

Because 30° and 60° are complementary angles — they add up to 90°. The co-function identities state:

sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)
tan(θ) = cot(90° - θ)

So: sin(30°) = cos(90° - 30°) = cos(60°) = 1/2 ✓

This makes intuitive sense from the 30-60-90 triangle. The side that is 'opposite' the 30° angle is 'adjacent' to the 60° angle, and vice versa. So sin(30°) and cos(60°) both equal 1/2 because they use the same side (length 1) and the same hypotenuse (length 2).

This is why the prefix 'co' in cosine, cotangent, and cosecant stands for 'complement' — cosine of an angle is the sine of its complement.

NAME	FORMULA	RESULT
Pythagorean identity at 30°	sin²(30°)+cos²(30°) = (1/2)²+(√3/2)² = 1/4+3/4	= 1 ✓
Tan from sin/cos	tan(30°) = sin(30°)/cos(30°) = (1/2)/(√3/2)	= 1/√3 = √3/3
Csc(30°)	csc(30°) = 1/sin(30°) = 1/(1/2)	= 2
Sec(30°)	sec(30°) = 1/cos(30°) = 1/(√3/2)	= 2/√3 = 2√3/3
Cot(30°)	cot(30°) = cos(30°)/sin(30°) = (√3/2)/(1/2)	= √3
Co-function identity	sin(30°) = cos(60°) = 1/2	Complementary angles sum to 90°
Double angle (sin)	sin(60°) = 2·sin(30°)·cos(30°) = 2·(1/2)·(√3/2)	= √3/2 ✓
30-60-90 triangle	Sides: 1, √3, 2 (ratio)	Short leg=1, long leg=√3, hypotenuse=2

Sin Cos Tan of 30 Degrees — Exact Values

TRIG VALUES AT 30°

UNIT CIRCLE

FORMULAS & IDENTITIES

WORKED EXAMPLE

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

MORE TRIGONOMETRY CALCULATORS

Function	Exact Value	Decimal
sin(30°)	1/2	0.5
cos(30°)	√3/2	0.8660254
tan(30°)	1/√3	0.57735027
csc(30°)	2	2
sec(30°)	2√3/3	1.15470054
cot(30°)	√3	1.73205081

°	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