The value of sin 120° is √3/2 ≈ 0.866. Since 120° lies in the second quadrant, where sine is positive, we use the identity sin(180° − θ) = sin θ to get sin 120° = sin(180° − 60°) = sin 60° = √3/2. The related values are cos 120° = −1/2 and tan 120° = −√3.
sin 120° = √3/2 ≈ 0.8660.
120° is in the second quadrant, where sine is positive.
sin 120° = sin(180° − 60°) = sin 60° = √3/2 (supplementary angle identity).
cos 120° = −1/2, tan 120° = −√3, sec 120° = −2.
sin 120° = sin 60° because 120° and 60° are supplementary angles.
On the unit circle, 120° corresponds to the point (−1/2, √3/2).
In radians: 120° = 2π/3, so sin(2π/3) = √3/2.
sin 120° = √3/2 ≈ 0.8660
Derivation using the supplementary angle identity: 120° = 180° − 60°
Using the identity: sin(180° − θ) = sin θ sin 120° = sin(180° − 60°) = sin 60° = √3/2
Since 120° is in the second quadrant (between 90° and 180°), and sine is positive in the second quadrant (S in the ASTC rule), sin 120° is positive.
Numerical value: sin 120° = √3/2 = 1.732.../2 ≈ 0.8660
The ASTC (All-Sin-Tan-Cos) rule, also remembered as 'All Students Take Calculus', tells us which trig functions are positive in each quadrant:
120° lies in Quadrant II, so:
This tells us the signs before we compute values: sin 120° = +sin 60° = +√3/2, cos 120° = −cos 60° = −1/2, tan 120° = −tan 60° = −√3.
Using the reference angle of 60° and the ASTC rule:
Reference angle for 120°: 180° − 120° = 60°
| Function | Value | Exact form |
|---|---|---|
| sin 120° | 0.8660 | √3/2 |
| cos 120° | −0.5 | −1/2 |
| tan 120° | −1.7321 | −√3 |
| cosec 120° | 1.1547 | 2/√3 = 2√3/3 |
| sec 120° | −2 | −2 |
| cot 120° | −0.5774 | −1/√3 = −√3/3 |
Verification using Pythagorean identity: sin²120° + cos²120° = (√3/2)² + (−1/2)² = 3/4 + 1/4 = 4/4 = 1 ✓
Verification using tan: tan 120° = sin 120°/cos 120° = (√3/2)/(−1/2) = −√3 ✓
Complete sine value table for standard angles:
| Angle | sin value | Exact form |
|---|---|---|
| 0° | 0 | 0 |
| 30° | 0.5 | 1/2 |
| 45° | 0.7071 | 1/√2 |
| 60° | 0.8660 | √3/2 |
| 90° | 1 | 1 |
| 120° | 0.8660 | √3/2 |
| 135° | 0.7071 | 1/√2 |
| 150° | 0.5 | 1/2 |
| 180° | 0 | 0 |
Notice the symmetry: sin 60° = sin 120° = √3/2. This is because 60° and 120° are supplementary angles (60° + 120° = 180°), and supplementary angles have equal sines.
Generally: sin θ = sin(180° − θ) for all angles θ. So: sin 30° = sin 150°, sin 45° = sin 135°, sin 60° = sin 120°.
On the unit circle (circle of radius 1 centred at origin), the angle 120° corresponds to the point:
(cos 120°, sin 120°) = (−1/2, √3/2)
This point is in the second quadrant (negative x, positive y), confirming:
The sine of any angle on the unit circle equals the y-coordinate of the corresponding point.
Alternate derivation using the unit circle:
Radians: 120° = 2π/3 radians. So sin(2π/3) = √3/2.
sin 120° = √3/2 ≈ 0.8660. Since 120° = 180° − 60°, and using the identity sin(180° − θ) = sin θ, we get sin 120° = sin 60° = √3/2.
120° is in the second quadrant (between 90° and 180°). In the second quadrant, only sine (and cosecant) are positive. Therefore sin 120° = +√3/2 is positive, while cos 120° = −1/2 and tan 120° = −√3 are negative.
cos 120° = −1/2 and tan 120° = −√3. Using the reference angle 60°: cos 120° = −cos 60° = −1/2 (negative in Q2) and tan 120° = −tan 60° = −√3 (negative in Q2).
On the unit circle, 120° corresponds to the point (cos 120°, sin 120°) = (−1/2, √3/2). The x-coordinate gives cosine and the y-coordinate gives sine.
120° = 2π/3 radians. So sin 120° = sin(2π/3) = √3/2 ≈ 0.8660.
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