Sine, cosine and tangent calculator, SOH CAH TOA with steps.

SOH CAH TOA is the memory aid for the three main trigonometric ratios of a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. This sine, cosine and tangent calculator gives all six ratios of an angle, solves a right triangle from any two known values, and turns a ratio back into an angle. It names the SOH, CAH or TOA rule it used, so you can follow the working.

How to use the sine, cosine and tangent calculator

1. Choose Angle to ratios, Right triangle or Ratio to angle.
2. For ratios, type the Angle and pick degrees, radians or gradians. Sin, cos, tan and their reciprocals csc, sec and cot appear, with exact values at angles such as 30°, 45° and 60°.
3. For a right triangle, fill in exactly two of Angle θ, Opposite, Adjacent, Hypotenuse and Area. Leave the others empty.
4. For the inverse, pick sin, cos, tan, csc, sec or cot, type the value, and read the angle in degrees, in radians and in the full turn.

The three ratios

Worked example: the ratios of 35°

sin 35° = 0.5736, cos 35° = 0.8192 and tan 35° = 0.7002. Check that tan = sin ÷ cos: 0.5736 ÷ 0.8192 = 0.7002. The reciprocals are csc 35° = 1.7434, sec 35° = 1.2208 and cot 35° = 1.4281.

Worked example: find the sides

A right triangle has θ = 30° and an opposite side of 5. The opposite and the hypotenuse go with sine, so sin 30° = 5 ÷ hypotenuse. That gives hypotenuse = 5 ÷ 0.5 = 10. The adjacent side uses tangent: adjacent = 5 ÷ tan 30° = 8.6603. Check with Pythagoras: 5² + 8.6603² = 25 + 75 = 100 = 10².

Worked example: a ladder

A 6 m ladder leans at 70° to the ground. The ladder is the hypotenuse, the height up the wall is the opposite side, and the distance from the wall is the adjacent side. Height = 6 × sin 70° = 5.6382 m and base = 6 × cos 70° = 2.0521 m. Enter angle 70 and hypotenuse 6 to see it.

Worked example: two sides to an angle

With opposite 3 and adjacent 4, tan θ = 3 ÷ 4 = 0.75, so θ = arctan 0.75 = 36.87° and the hypotenuse is √(9 + 16) = 5. The other acute angle is 90° − 36.87° = 53.13°.

Worked example: from the area

A right triangle has an area of 24 and an adjacent side of 6. Since area = ½ × opposite × adjacent, the opposite side is 2 × 24 ÷ 6 = 8. Then tan θ = 8 ÷ 6, so θ = 53.13° and the hypotenuse is √(64 + 36) = 10. An area with only the hypotenuse is not enough, because two different triangles fit it, so the page asks for one more value.

From a ratio back to an angle

If sin θ = 0.5, then θ = arcsin 0.5 = 30°. The same sine also happens at 150° in one full turn, because sine is positive in two quadrants. The page lists both. A sine or cosine can only be from −1 to 1, so a value such as 1.5 has no angle.

Exact values at the special angles

Some angles have clean answers that come from half a square or half an equilateral triangle. The calculator shows these exact forms next to the decimals. They are worth knowing by heart for exams, because most calculator-free questions use them.

Look at the sine column going down: 0, 1/2, √2/2, √3/2, 1. Written as √0/2, √1/2, √2/2, √3/2, √4/2 it is easy to remember, and the cosine column is the same list upside down. This works because the cosine of an angle equals the sine of its complement, so cos 30° = sin 60°.

Angles bigger than 90 degrees

The right triangle picture only holds from 0° to 90°, but the ratios exist for every angle. The trick is the reference angle, the acute angle between the arm of the angle and the x axis. For 120° that is 180° − 120° = 60°, so sin 120° = sin 60° = √3/2, cos 120° = −cos 60° = −1/2 and tan 120° = −√3. For 225° the reference angle is 45°, so sin 225° = cos 225° = −√2/2 and tan 225° = 1. The page gives the quadrant, the reference angle and a unit circle drawing with the cosine and sine marked.

Negative angles turn clockwise, so −45° lands in quadrant IV: sin(−45°) = −√2/2 and cos(−45°) = √2/2. Angles over 360° wrap round: sin 720° = sin 0° = 0.

Exact values at 15 and 75 degrees

Besides 30°, 45° and 60°, the page gives exact values for 15° and 75° and their mirror angles. sin 15° = (√6 − √2) ÷ 4 = 0.2588, cos 15° = (√6 + √2) ÷ 4 = 0.9659 and tan 15° = 2 − √3 = 0.2679. At 75° the sine and cosine swap and tan 75° = 2 + √3 = 3.7321. The reciprocals csc, sec and cot are shown exactly at 30°, 45° and 60° and their mirror angles, for example sec 30° = 2√3/3.

Degrees, minutes and seconds

Surveying and navigation angles come as 35° 30' 15". Type it that way. The page turns it into 35 + 30 ÷ 60 + 15 ÷ 3600 = 35.504167° and gives sin = 0.5808, cos = 0.8141 and tan = 0.7134. Every angle result also shows its degrees, minutes and seconds form.

Inverse csc, sec and cot

These are worked through the ratio they are the reciprocal of. If sec θ = −2 then cos θ = −1/2, so θ = 120° or 240° in one full turn. If csc θ = 2 then sin θ = 1/2, so θ = 30° or 150°. A csc or sec can never be between −1 and 1, and the page says so. The calculator shows the principal value, which is −90° to 90° for arcsin and arctan and 0° to 180° for arccos, then lists the angles in 0° to 360°. For sin θ = −0.5 the principal value is −30°, and in the full turn it is 210° and 330°.

Using the ratios in real problems

Heights and distances are the classic uses. To find the height of a tree, stand a known distance away, measure the angle up to the top, and use tan θ = height ÷ distance. A tree seen at 40° from 12 m away is 12 × tan 40° = 10.07 m tall (plus your eye height). Slopes use the same idea: a ramp that rises 1 for every 12 along is at arctan(1/12) = 4.76°. For roof pitch, stairs and cables, the hypotenuse is the length of material you need and the two legs are the rise and the run.

Common problems

• Using the right triangle for a big angle. A right triangle only has angles under 90°. For 120° or 225° use the reference angle and the signs in the table above.
• Wrong angle mode. sin 30 gives 0.5 in degrees but −0.988 if your tool is set to radians. Check the unit box.
• Labelling the sides from the wrong corner. Opposite and adjacent depend on which angle you call θ. The hypotenuse is always across from the right angle.
• A hypotenuse that is too short. It must be the longest side. The page tells you when it is not.

For angles beyond 90°, see the unit circle chart. For triangles without a right angle use the law of cosines calculator or the triangle calculator.