Use two triangles for special-angle values and the unit circle for signs. This guide is for learners rebuilding a forgotten value, choosing the sign at 150°, or checking degree/radian inputs. Check the unit before calculating.
In the trigonometry calculator, select degrees and enter 150 to compare sin=1/2, cos=−√3/2 and tan=−√3/3 with the diagram.
Remember sin, cos and tan by naming the sides
For the chosen acute angle θ in a right triangle, the hypotenuse is opposite the right angle, the opposite side faces θ, and the adjacent side touches θ but is not the hypotenuse. Choosing the other acute angle swaps opposite and adjacent.
| Function | Acute right-triangle ratio | Unit-circle definition | Domain condition |
|---|---|---|---|
| sinθ | Opposite/hypotenuse | Vertical coordinate y | Every real angle |
| cosθ | Adjacent/hypotenuse | Horizontal coordinate x | Every real angle |
| tanθ | Opposite/adjacent | y/x=sinθ/cosθ | cosθ≠0 |
Derive 30° and 60° from 1:√3:2; 45° from 1:1:√2
Halve an equilateral triangle of side 2. The half-base is 1 and the height is √(2²−1²)=√3. Looking from its 30° angle, the opposite side is 1, giving sin30°=1/2. From 60°, the opposite side is √3, giving sin60°=√3/2. In a right isosceles triangle with legs 1 and 1, the hypotenuse is √2, so sin45°=cos45°=1/√2=√2/2.
Exact values for 0°, 30°, 45°, 60° and 90°
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | Undefined |
Use unit-circle points (1,0) and (0,1) for 0° and 90°, rather than dividing sides in a collapsed triangle. tan90° is undefined because it would divide 1 by zero. The reciprocal of a tiny rounded cosine is not an answer for tan90°.
Convert degrees and radians before entering values
Radian measure is arc length s divided by radius r. A half-circle gives πr/r=π, so 180°=π rad. Multiply degrees by π/180; multiply radians by 180/π.
| Given angle | Conversion | Other unit | Input mode |
|---|---|---|---|
| 30° | 30×π/180 | π/6 rad | Degrees, 30 |
| 150° | 150×π/180 | 5π/6 rad | Degrees, 150 |
| −30° | −30×π/180 | −π/6 rad | Degrees, −30 |
| 2π rad | 2π×180/π | 360° | Radians, 2pi |
| π/6 rad | (π/6)×180/π | 30° | Radians, pi/6 |
| 30 rad | 30×180/π | About 1718.873385° | Radians, 30 |
sin30°=0.5, but sin(30 rad)≈−0.988032. Unit buttons do not convert the input number. Switching an unchanged 30 to radians calculates 30 rad. The π/6 example button sets both radians and the example value.
Check the conditions, then calculate
Check that 30° and π/6 agree
Select the 30° example, then the π/6 example. Compare sine 1/2, cosine √3/2 and tangent √3/3, including the unit display.
Signs and reference angles: 150°, 225° and −30°
The unit-circle point is P=(cosθ,sinθ). Positive rotation is counterclockwise from 0°. Axis angles are excluded from the open quadrant intervals below.
| Quadrant in one turn | sin | cos | tan | Reference-angle example |
|---|---|---|---|---|
| I: 0°<θ<90° | + | + | + | 30°→30° |
| II: 90°<θ<180° | + | − | − | 150°→180−150=30° |
| III: 180°<θ<270° | − | − | + | 225°→225−180=45° |
| IV: 270°<θ<360° | − | + | − | 330°→360−330=30° |
At 150°, quadrant II gives sin=1/2, cos=−√3/2 and tan=−√3/3. At 225°, sine and cosine are both −√2/2, so tangent is 1.
−30° points in the same direction as 330°, giving sin=−1/2, cos=√3/2 and tan=−√3/3. This does not make −30°=330° a unit-conversion equality. They differ by a full turn. Sine and cosine have period 360° and tangent has period 180°. A sine value alone does not recover a unique original angle.
Applied example: a slope’s length is not its horizontal run
A 10 m slope at 30° above horizontal has height 10sin30°=5 m and horizontal run 10cos30°=5√3 m, about 8.660254 m. If 10 m instead describes the horizontal run, the height is 10tan30°≈5.773503 m. Label the hypotenuse and adjacent side before choosing a function. See the vector guide for components.
Tool syntax, approximation and unsupported tasks
Enter a number or a pi expression such as pi/6, 5pi/6 or 2pi. The absolute input limit is one million in the selected unit, after evaluating a pi expression: one million degrees in degree mode, or one million radians in radian mode. Enter only an angle; expressions such as sin(30) and 30+60 are not parsed.
Function values and quadrants use a representative within one turn. The other-unit display converts the original rotation: −30° gives about −0.523599 rad, and 2π rad gives 360°. Those are distinct from the representative angles 330° and 0°.
Separate symbolic fractions/radicals at special angles from decimal approximations. Ordinary decimals show at most six places; small nonzero values use scientific notation. Near boundaries such as 90°, rounding is sensitive. The tool treats tangent as undefined when the computed absolute cosine is below 10⁻¹². This numerical rule includes a narrow neighborhood of the mathematically undefined angle. This is not a limit solver or a measurement-uncertainty evaluator.
Read the result in order: unit, quadrant/sign, symbolic or approximate value, then any division by zero. Decimal rounding cannot correct a wrong unit.
Verify, copy and return later
At 150°, sin²+cos²=(1/2)²+(−√3/2)²=1 and tan=sin/cos=−√3/3. That identity alone cannot detect a wrong unit; also check the known 30° value.
Check the conditions, then calculate
Check the signs as well as the values
Enter 150 in degree mode and inspect quadrant II, reference angle 30° and all three values. Add “150°” to your notes beside the copied results.
Favorite the tool for a shortcut in the same browser. This saves neither input history nor notes and does not synchronize devices. Return through the mathematics category or compare logarithms and permutations and combinations.
Editorial verification scope
On October 29, 2026, Tools Hub Editorial Team checked the public teaching sources and the special-angle, sign and conversion examples. The diagrams are original constructions from side ratios and unit-circle coordinates. Local browser verification covers examples, negative angles, multiple turns, errors and copy.
FAQ
What are sin30°, cos30° and tan30°?
They are 1/2, √3/2 and √3/3. Use a right triangle with opposite side 1, adjacent side √3 and hypotenuse 2 relative to the 30° angle.
Why does pi/6 not give sine 0.5?
Select radians first. pi/6 radians is 30°, but pi/6 in degree mode is about 0.523599°. Unit buttons reinterpret the input; they do not convert the number in the input field.
Is tan90° infinity?
It is undefined as a real value because tan=sin/cos and cos90°=0. Values near 90° become large with different signs on either side, but that does not assign infinity to tan90°.
Are −30° and 330° the same angle?
Their rotations differ by 360°, but their terminal sides and trigonometric values agree. Unit conversion keeps −30°=−π/6 separate from 330°=11π/6. Sine and cosine have period 360°; tangent has period 180°.
