Sin Cos Tan Calculator
- The angle is 30°, which is π/6 radians.
- Find the quadrant and the reference angle: Quadrant I, reference angle 30°.
- Signs in Quadrant I: sine is positive, cosine is positive, tangent is positive.
- Read the values: sin = 1/2 ≈ 0.5, cos = √3/2 ≈ 0.8660254.
- Tangent is sin ÷ cos = √3/3 ≈ 0.5773503.
- Unit circle: for an angle θ, the point on the circle of radius 1 has x = cos θ and y = sin θ.
- Reference angle: any angle is matched to an angle between 0° and 90° with the same values apart from the sign.
- Signs by quadrant: sine is positive in quadrants I and II, cosine in I and IV, and tangent in I and III.
- Special angles: multiples of 15° have exact values built from √2, √3 and √6.
- Tangent: tan θ = sin θ ÷ cos θ, and it is undefined where cos θ = 0.
- Reciprocal functions: csc = 1/sin, sec = 1/cos and cot = 1/tan.
- sin² + cos² = 0.25 + 0.75 = 1. This is always 1.
- The ratio sin ÷ cos = 0.5 ÷ 0.8660254 = 0.5773503.
- 30° = π/6 radians, so the two units agree.
Sin Cos Tan Calculator: Sine, Cosine and Tangent of Any Angle
This free sin cos tan calculator gives the sine, cosine and tangent of an angle in degrees or radians. It also shows cosecant, secant and cotangent, the exact value when there is one, the quadrant and reference angle, and a unit circle that moves as you change the angle. Each step is written out, so it is as useful for checking homework as for quick answers.
Quick answer: for 30°, sin = 1/2 = 0.5, cos = √3/2 ≈ 0.866 and tan = √3/3 ≈ 0.577.
What Are Sine, Cosine and Tangent?
Sine, cosine and tangent link the angles of a right triangle to the lengths of its sides. Sine is the side opposite the angle divided by the hypotenuse, cosine is the adjacent side divided by the hypotenuse, and tangent is the opposite side divided by the adjacent side. A common way to remember this is SOH CAH TOA. The same functions work for any angle, not only those inside a triangle, when they are read from the unit circle.
The Unit Circle
The unit circle has a radius of 1 and is centred at the origin. Turn a line from the positive x-axis by an angle and it meets the circle at a point. The x-coordinate of that point is the cosine of the angle and the y-coordinate is the sine. Tangent is the height where the line, extended, meets the vertical line at x = 1. The graph above draws all three, so you can drag the slider and watch the values change.
Degrees and Radians
A full turn is 360 degrees or 2π radians, which means 180° = π radians. Degrees are common in everyday work and geometry, while radians are used in calculus, physics and most programming languages. Use the Degrees and Radians buttons to switch, and the box keeps the same angle by converting it for you. The angle in the other unit is always shown in the results.
Exact Values of Special Angles
For multiples of 15°, the calculator shows the exact value as well as the decimal. The values for 30°, 45° and 60° come from two familiar triangles and are worth learning. Some of the most used are listed below.
| Angle | sin | cos | tan |
|---|---|---|---|
| 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 |
| 180° (π) | 0 | −1 | 0 |
| 270° (3π/2) | −1 | 0 | undefined |
| 360° (2π) | 0 | 1 | 0 |
Quadrants and Signs
The sign of each function depends on the quadrant the angle ends in. In the first quadrant, from 0° to 90°, all three are positive. In the second, only sine is positive, in the third only tangent, and in the fourth only cosine. Every angle also has a reference angle between 0° and 90°, the acute angle it makes with the x-axis, which has the same values apart from the sign. The calculator shows both.
Reciprocal Functions
Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent. They are found by dividing 1 by the original value, so they are undefined wherever that value is 0. For triangle work, the Pythagorean theorem calculator finds a missing side, and the circle calculator and the derivative calculator cover related topics. The scientific calculator and the math solver help with the arithmetic around it.
How to Use This Sin Cos Tan Calculator
Choose the unit
Pick Degrees or Radians so the angle is read correctly.
Enter the angle
Type it, drag the slider, or tap one of the common angles.
Read the results
See sine, cosine and tangent, with exact values for special angles.
Check the circle
Watch the unit circle and follow the working step by step.