Question 1

What is sin(60°)?

Accepted Answer

sin(60°) = √3/2 ≈ 0.86602540.

This comes from the 30-60-90 triangle with sides 1, √3, 2. At the 60° angle:
- Opposite side = √3
- Hypotenuse = 2
- sin = opposite/hypotenuse = √3/2

On the unit circle, 60° is the point (1/2, √3/2). The y-coordinate gives sin(60°) = √3/2.

Key relationship: sin(60°) = cos(30°) = √3/2. These two angles are complementary (sum to 90°), so their sine and cosine values swap.

Question 2

What is cos(60°)?

Accepted Answer

cos(60°) = 1/2 = 0.5.

From the 30-60-90 triangle at the 60° angle:
- Adjacent side = 1
- Hypotenuse = 2
- cos = adjacent/hypotenuse = 1/2

On the unit circle, 60° is the point (1/2, √3/2). The x-coordinate gives cos(60°) = 1/2.

Key relationship: cos(60°) = sin(30°) = 1/2. This is one of the simplest exact trig values — cos(60°) is exactly 0.5, making 60° one of the easiest angles to remember.

Question 3

What is tan(60°)?

Accepted Answer

tan(60°) = √3 ≈ 1.73205081.

tan = sin/cos = (√3/2)/(1/2) = √3

Alternatively: in the 30-60-90 triangle at the 60° angle, tan = opposite/adjacent = √3/1 = √3.

The pattern for tan at 30°, 45°, 60°: 1/√3, 1, √3. Each is the square root of the next: (1/√3)² = 1/3, 1² = 1, (√3)² = 3. This is a clean geometric progression.

tan(60°) > 1 because at 60° the opposite side (√3) is longer than the adjacent side (1), meaning the angle is steep — more vertical than horizontal.

Question 4

How are the trig values at 60° related to those at 30°?

Accepted Answer

The values at 60° and 30° are mirrors of each other — every function at 60° equals the co-function at 30°:

- sin(60°) = cos(30°) = √3/2
- cos(60°) = sin(30°) = 1/2
- tan(60°) = cot(30°) = √3
- cot(60°) = tan(30°) = 1/√3
- sec(60°) = csc(30°) = 2
- csc(60°) = sec(30°) = 2√3/3

This follows directly from the co-function identities: sin(θ) = cos(90°-θ), tan(θ) = cot(90°-θ), sec(θ) = csc(90°-θ).

Practical memory trick: if you know all six values at 30°, you automatically know all six at 60° by swapping sin↔cos, tan↔cot, sec↔csc.

Question 5

What is the 30-60-90 triangle and how does it give trig values at 60°?

Accepted Answer

A 30-60-90 triangle has angles 30°, 60°, and 90°, with sides in the ratio 1 : √3 : 2.

Origin: start with an equilateral triangle (all sides = 2, all angles = 60°). Draw the altitude from one vertex to the opposite side. This bisects both the angle (giving 30°) and the base (giving 1). The altitude itself has length √3 (by Pythagoras: 2² - 1² = 3, so √3).

At the 60° angle:
- Opposite = √3 (the altitude)
- Adjacent = 1 (half the base)
- Hypotenuse = 2 (original side)

So: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3

Practical uses of 60°:
- Equilateral triangles (all angles = 60°)
- Hexagonal grids and honeycomb structures
- 60° is the angle of incidence for total internal reflection in some optical fibres
- Clock hands form a 60° angle at 2 o'clock and 10 o'clock

°	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

NAME	FORMULA	RESULT
Pythagorean identity at 60°	sin²(60°)+cos²(60°) = (√3/2)²+(1/2)² = 3/4+1/4	= 1 ✓
Tan from sin/cos	tan(60°) = sin(60°)/cos(60°) = (√3/2)/(1/2)	= √3
Csc(60°)	csc(60°) = 1/sin(60°) = 1/(√3/2)	= 2/√3 = 2√3/3
Sec(60°)	sec(60°) = 1/cos(60°) = 1/(1/2)	= 2
Cot(60°)	cot(60°) = cos(60°)/sin(60°) = (1/2)/(√3/2)	= 1/√3 = √3/3
Mirror of 30°	sin(60°) = cos(30°) = √3/2	Co-function identity: sin(θ) = cos(90°-θ)
30-60-90 triangle at 60°	opposite=√3, adjacent=1, hypotenuse=2	tan(60°) = √3/1 = √3
Double angle (cos)	cos(120°) = 1-2sin²(60°) = 1-2(3/4)	= 1-3/2 = -1/2 ✓

Sin Cos Tan of 60 Degrees — Exact Values

TRIG VALUES AT 60°

UNIT CIRCLE

FORMULAS & IDENTITIES

WORKED EXAMPLE

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

MORE TRIGONOMETRY CALCULATORS

Function	Exact Value	Decimal
sin(60°)	√3/2	0.8660254
cos(60°)	1/2	0.5
tan(60°)	√3	1.73205081
csc(60°)	2√3/3	1.15470054
sec(60°)	2	2
cot(60°)	1/√3	0.57735027