Trigonometry Calculator
A trigonometry calculator for sin, cos and tan, plus csc, sec and cot, with the exact value at the special angles (sin 60° is √3/2) and inverse trig from a value back to every angle that produces it.
sin, cos, tan of 150°
- sin(150°) The y-coordinate of the point on the unit circle. Reference angle 30° gives 1/2, and sine is positive in quadrant II.
- 1/2 = 0.5
- cos(150°) The x-coordinate. Reference angle 30° gives √3/2, and cosine is negative in quadrant II.
- -√3/2 ≈ -0.8660254038
- tan(150°) sin / cos: the slope of the line from the origin to the point.
- -√3/3 ≈ -0.5773502692
- csc(150°) 1 / sin.
- 2
- sec(150°) 1 / cos.
- -2√3/3 ≈ -1.154700538
- cot(150°) cos / sin, or 1 / tan.
- -√3 ≈ -1.732050808
- Point on the unit circle (cos θ, sin θ), on a circle of radius 1, so sin² + cos² = 1.
- (-√3/2, 1/2)
- Quadrant Quadrant II: only sine and its reciprocal cosecant are positive (the S in ASTC). x is negative here, so cosine is.
- II
- Reference angle The acute angle to the x-axis: 180° - θ.
- 30°
- In degrees
- 150°
- In radians
- 5π/6 ≈ 2.617993878
- In gradians
- 166.6666667 grad
| Angle | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| 90° (π/2) | 1 | 0 | undefined (cos 0) | 1 | undefined (cos 0) | 0 |
| 120° (2π/3) | √3/2 | -1/2 | -√3 | 2√3/3 | -2 | -√3/3 |
| 135° (3π/4) | √2/2 | -√2/2 | -1 | √2 | -√2 | -1 |
| 150° (5π/6) | 1/2 | -√3/2 | -√3/3 | 2 | -2√3/3 | -√3 |
| 180° (π) | 0 | -1 | 0 | undefined (sin 0) | -1 | undefined (sin 0) |
How the Trigonometry Calculator works
The six trigonometric functions all come from one picture: a circle of radius 1 centered on the origin, and a line from the center at angle θ, measured counterclockwise from the positive x-axis. Where that line meets the circle is the point (cos θ, sin θ). Cosine is the x-coordinate, sine is the y-coordinate, and tangent is their ratio, sin θ / cos θ, which is the slope of the line. The other three are reciprocals: cosecant is 1/sin, secant is 1/cos and cotangent is 1/tan.
In a right triangle the same definitions read as the familiar SOH CAH TOA: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. The unit circle is what extends those ratios past 90 degrees, where a right triangle stops making sense but a point on a circle does not.
EXACT VALUES AT THE SPECIAL ANGLES. At 30, 45 and 60 degrees the values come from two triangles everyone can draw. Half a square, cut along the diagonal, has angles of 45 degrees and sides 1, 1 and √2, so sin 45° = cos 45° = 1/√2 = √2/2. Half an equilateral triangle has angles of 30 and 60 degrees and sides 1, √3 and 2, so sin 30° = 1/2, sin 60° = √3/2 and tan 60° = √3. That is the whole table. Everything at 120, 135, 150, 210 degrees and so on is one of those values with a sign attached.
THE REFERENCE ANGLE AND THE SIGN. The reference angle is the acute angle between the line and the x-axis: θ itself in quadrant I, 180° − θ in quadrant II, θ − 180° in quadrant III and 360° − θ in quadrant IV. The functions of θ have the same size as the functions of the reference angle, and the quadrant decides the sign. The mnemonic is ASTC, read counterclockwise from quadrant I as All, Sine, Tangent, Cosine: the functions that are positive in each quadrant. So sin 150° is positive and equal to sin 30° = 1/2, while cos 150° is negative: −√3/2. Getting the size right and the sign wrong is the commonest error in trigonometry homework.
UNDEFINED, NOT HUGE. Tangent is sin/cos, so wherever cosine is zero (90°, 270°, and every half-turn from them), tangent is undefined: the line is vertical and has no slope. Secant is undefined at the same angles, and cosecant and cotangent wherever sine is zero.
DEGREES, RADIANS AND GRADIANS. A full turn is 360 degrees, 2π radians or 400 gradians. Radians are the natural unit (an angle of one radian cuts off an arc equal to the radius), which is why calculus uses them. To convert, multiply degrees by π/180. Gradians, 100 to a right angle, are mostly used in surveying.
INVERSE FUNCTIONS run the other way: arcsin(1/2) asks which angle has a sine of 1/2. Because sine repeats, infinitely many do, so each inverse is defined to return one principal value: arcsin and arctan between −90° and 90°, arccos between 0° and 180°. In one full turn there are usually two answers: arcsin(1/2) is 30°, and 150° also has a sine of 1/2. Equations like sin θ = 1/2 on an exam want both, and the second comes from the symmetry of the circle: θ and 180° − θ for sine, θ and 360° − θ for cosine, θ and θ + 180° for tangent.
To solve a whole triangle from some of its sides and angles, use the law of sines and the law of cosines on the triangle calculator.
Trigonometry Calculator FAQ
What is sin 60 degrees exactly?
√3/2, about 0.8660254. It comes from half an equilateral triangle with sides 1, √3 and 2: the side opposite the 60° angle is √3 and the hypotenuse is 2. cos 60° is 1/2 and tan 60° is √3.
Why is tan 90 degrees undefined?
Tangent is sin/cos, and cos 90° is exactly 0, so the division has no answer. Geometrically the line at 90° is vertical and a vertical line has no slope.
How do I find the reference angle?
It is the acute angle to the x-axis. In quadrant I it is the angle itself; in quadrant II, 180° minus the angle; in quadrant III, the angle minus 180°; in quadrant IV, 360° minus the angle. 150° has a reference angle of 30°, so its functions have the size of 30°'s, signed for quadrant II.
What does ASTC mean?
All, Sine, Tangent, Cosine: the functions that are positive in quadrants I, II, III and IV in turn. In quadrant II only sine (and cosecant) is positive, so sin 150° = 1/2 but cos 150° = -√3/2. Some people remember it as All Students Take Calculus.
Why does my calculator say sin 30 = -0.988?
It is in radian mode. 30 radians is nearly five full turns and lands at an angle whose sine is about -0.988. In degree mode sin 30° is exactly 1/2.