All Solvers/Trigonometry Calculator
Degrees & Radians, 6 Functions

Trigonometry Calculator sin · cos · tan

Evaluate sin, cos, tan and their inverses for any angle, in degrees or radians, with step-by-step explanations and common special values.

Trigonometric function

sin(x)

Result

sin(30°) = 0.5

Step 1. Convert to radians: 30° × π/180 = 0.52359878 rad

Step 2. Evaluate sin at this value: 0.5

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Trig functions at a glance

Unit Circle Definitions

sin(θ) = y   cos(θ) = x   tan(θ) = y/x

On the unit circle (radius 1), the point at angle θ has coordinates (cos θ, sin θ). Tangent is their ratio.

The three primary trigonometric functions, sine, cosine, and tangent, relate the angles of a right triangle to the ratios of its sides. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

For arbitrary angles, the functions are defined via the unit circle. Calculator inputs accept both degrees (e.g. 45°) and radians (e.g. π/4). Inverse functions (arcsin, arccos, arctan) take a ratio and return the angle.

Worked Examples

Five trigonometry computations, from special angles to inverse

Special-value angles you should memorize, plus inverse functions and angle-sum decomposition.

Example 1, sin(30°)

Find sin(30°)

0.5
  1. 1

    Convert to radians (optional)

    30° × π/180 = π/6 rad
  2. 2

    Recall the special value

    sin(30°) = sin(π/6) = 1/2
  3. 3

    Result

    0.5

30°, 45°, 60° give exact values you should memorize.

Example 2, cos(60°)

Find cos(60°)

0.5
  1. 1

    Special angle

    cos(60°) = cos(π/3)
  2. 2

    Exact value

    cos(π/3) = 1/2
  3. 3

    Result

    0.5

Cosine of complement: cos(60°) = sin(30°) = 0.5.

Example 3, tan(45°)

Find tan(45°)

1
  1. 1

    Identity

    tan(θ) = sin(θ) / cos(θ)
  2. 2

    Substitute

    tan(45°) = sin(45°) / cos(45°) = (√2/2) / (√2/2)
  3. 3

    Result

    1

When sine and cosine are equal, tangent is 1.

Example 4, Inverse sine

Find sin⁻¹(0.5)

30°
  1. 1

    Question

    What angle has sin = 0.5?
  2. 2

    Recall

    sin(30°) = 0.5
  3. 3

    Result

    sin⁻¹(0.5) = 30° (or π/6 rad)

Inverse trig functions return angles. Domain is restricted, sin⁻¹ ∈ [−90°, 90°].

Example 5, Trig at non-standard angle

Find sin(75°)

≈ 0.9659
  1. 1

    Decompose

    sin(75°) = sin(45° + 30°)
  2. 2

    Sum formula

    = sin(45°)cos(30°) + cos(45°)sin(30°)
  3. 3

    Substitute

    = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
  4. 4

    Result

    ≈ 0.9659

For angles outside the special set, use the calculator, you can also derive via sum formulas.

Free online trigonometry calculator with step-by-step solutions

Whether you are learning trig for the first time, prepping for the SAT or ACT, or working through an engineering problem, this calculator gives you sine, cosine, and tangent values quickly, in both degrees and radians, with the full reasoning shown. Inverse functions (arcsin, arccos, arctan) are also supported.

Sine, cosine, and tangent explained

In a right triangle with hypotenuse h, opposite side o, and adjacent side a, the sine of the angle is o/h, cosine is a/h, and tangent is o/a. On the unit circle (radius 1), the x-coordinate of a point is cos(θ) and the y-coordinate is sin(θ), tan(θ) is the slope of the line from origin to that point.

Degrees and radians

Degrees split a circle into 360 parts, radians split it into 2π parts. To convert: degrees × π/180 = radians; radians × 180/π = degrees. Calculus and higher math almost always use radians, applied geometry and engineering often use degrees. This calculator handles both seamlessly.

Inverse trigonometric functions

The inverse functions, sin⁻¹, cos⁻¹, tan⁻¹ (also called arcsin, arccos, arctan), take a ratio and return the angle that produces it. Because trig functions are periodic, the inverses return a value within a restricted range. For sin⁻¹, the output is between −90° and 90°. For cos⁻¹, between 0° and 180°. For tan⁻¹, strictly between −90° and 90°.

Special values worth memorizing

Sin and cos of 0°, 30°, 45°, 60°, and 90° are exact values that appear constantly: 0, 1/2, √2/2, √3/2, and 1 respectively. Knowing these by heart speeds up algebra and lets you spot when a calculator answer looks wrong. The calculator above will help build that intuition by showing how exact values relate to decimal approximations.

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