Trigonometry in MV Workstation
Sine, cosine, tangent, their reciprocals and inverses. Angles can be written in degrees, radians or turns.
Type the maths the way you write it and MV Workstation shows the exact answer, the working and, where it helps, a picture you can drag.
Things to try
| What it is | Type this |
|---|---|
| Right angle | sin(pi/2) |
| Degrees | sin 30 degrees |
| Reciprocal | sec 60° |
| In plain English | the sine of 30 degrees |
| Identity | sin^2(x) + cos^2(x) |
| Inverse function | sin^-1(1) |
Tools
- Exact value: The exact value of sin, cos, tan, sec, csc, cot (or an inverse) at a multiple of 15°, with the quadrant, reference angle and CAST rule shown.
- Convert an angle: Write an angle in degrees, radians, turns and degrees-minutes-seconds at once: 150° is 5π/6 rad.
- Check an identity: Prove that two trig expressions are always equal, or find a value where they are not.
Pictures you can drag
- Right-angled triangle (SOH CAH TOA): Opposite, adjacent and hypotenuse for the angle you choose, with the ratios worked out. Drag the corner.
- Special triangles: The 45-45-90 and 30-60-90 triangles, where the exact values come from.
- The unit circle: Drag the point round: sin, cos and tan appear as lengths, with exact values, the CAST quadrants and Play.
- Radians, arc length and sector area: See one radian, then arc length, sector and segment area for any angle.
- From the circle to the wave: Watch a point going round the circle draw the sine, cosine or tangent wave.
- Angle measures: The same angle in degrees, minutes and seconds, radians, turns and gradians.
- Table of trig values: Exact values at 15°, 30°, 45° or 90° steps; click a row to highlight it.
- Angles of elevation and depression: A tower, a cliff, or two observation points: find the height from an angle.
- Similar triangles: Enlarge a triangle and see the sides scale while the angles stay the same.
- The sine rule: Reshape any triangle and see a/sin A = b/sin B = c/sin C, with the circle through the corners.
- The cosine rule: Reshape a triangle; Pythagoras gets corrected by the cosine term.
- Area of a triangle: ½ab sin C: Half base times height and ½ab sin C side by side, plus how the area depends on C.
- The ambiguous case (SSA): Zero, one or two triangles from two sides and a non-included angle.
- Triangle solver: Tell it three measurements (SSS, SAS, ASA, AAS, SSA, right-angled) and it finds the rest, with the working.
- Trig graph explorer: y = a·f(b(x − c)) + d: slide, stretch and squeeze a sine, cosine or tangent graph by dragging.
- Inverse trig functions: arcsin, arccos and arctan as reflected graphs, with the one answer they allow.
- Solving a trig equation: Move the line y = k and see every solution, plus the general solution.
- Is it an identity?: Plot both sides of a statement and test it at hundreds of angles.
- The R-form: a sin x + b cos x: Two waves add into one wave R sin(x + α), shown with a phasor.
- Trigonometry in 3D: A box with its diagonals: angles between a line and a plane, found from right triangles.
Functions
| Function | What it does | Example |
|---|---|---|
sin(x) | Sine of x. Plain numbers and pi are radians; write 30 degrees or 30° for degrees. | sin(pi/2) |
cos(x) | Cosine of x, in radians. | cos(0) |
tan(x) | Tangent of x, in radians. | tan(pi/4) |
asin(x) | Inverse sine, result in radians. Also written sin^-1(x). | asin(1) |
acos(x) | Inverse cosine, result in radians. Also written cos^-1(x). | acos(0) |
atan(x) | Inverse tangent, result in radians. Also written tan^-1(x). | atan(1) |
sec(x) | Secant: 1 / cos(x). | sec(pi/3) |
csc(x) | Cosecant: 1 / sin(x). Also written cosec. | csc(pi/6) |
cot(x) | Cotangent: 1 / tan(x). | cot(pi/4) |
asec(x) | Inverse secant. Also written arcsec or sec^-1(x). | asec(2) |
acsc(x) | Inverse cosecant. Also written arccsc or csc^-1(x). | acsc(2) |
acot(x) | Inverse cotangent, between 0 and pi. Also written arccot. | acot(1) |
sinh(x) | Hyperbolic sine. | sinh(1) |
cosh(x) | Hyperbolic cosine. | cosh(1) |
tanh(x) | Hyperbolic tangent. | tanh(1) |
Open MV Workstation, it is free
MV Workstation is a visual maths whiteboard that solves, shows the steps and draws as you type. It runs in your browser and works offline.