Trigonometry

Trigonometry Solver

Compute sine, cosine, tangent and their inverses with degrees or radians.

Common Angle Reference

DegreesRadianssincostan
0°0010
30°π/61/2√3/21/√3
45°π/4√2/2√2/21
60°π/3√3/21/2√3
90°π/210∞
180°π0−10
270°3π/2−10∞
360°2π010

Understanding Trigonometry

Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles. It is indispensable in fields ranging from architecture and civil engineering to astronomy, navigation, and computer graphics. TheCalcPro's trigonometry solver computes all six trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — in both degree and radian mode, with instant results and no server-side processing.

Core Trigonometric Ratios (Right Triangle)

In a right triangle with an acute angle θ, the three primary ratios are defined as:

sin(θ) = Opposite / Hypotenuse

cos(θ) = Adjacent / Hypotenuse

tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ)

The reciprocal functions are: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = 1/tan(θ). Together, these six functions completely describe the angular relationships in any triangle.

Step-by-Step Example: Finding the Height of a Building

You stand 30 meters from the base of a building. Using a clinometer, you measure the angle of elevation to the rooftop as 55°. How tall is the building?

  1. Identify the triangle: The distance from you to the building is the adjacent side (30 m), and the height is the opposite side (unknown h).
  2. Choose the right ratio: tan(θ) = Opposite / Adjacent, so tan(55°) = h / 30
  3. Compute tan(55°): tan(55°) = 1.4281
  4. Solve for h: h = 30 × 1.4281 = 42.84 meters

The building is approximately 42.84 meters tall. This surveying technique is used by architects, land surveyors, and construction professionals worldwide. For more advanced function analysis, explore our Function Plotter to visualize trig functions graphically.

Trigonometry FAQ

SOH-CAH-TOA is a mnemonic for the three primary trigonometric ratios in a right triangle. SOH: Sine = Opposite / Hypotenuse. CAH: Cosine = Adjacent / Hypotenuse. TOA: Tangent = Opposite / Adjacent. These ratios relate the sides of a right triangle to its acute angles and form the foundation of all trigonometric calculations.

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Common conversions: 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. The radian is the standard unit in calculus and physics because it simplifies many formulas (e.g., arc length s = rθ works directly in radians).

Inverse trigonometric functions (arcsin, arccos, arctan) reverse the standard trig functions. If sin(θ) = 0.5, then arcsin(0.5) = 30° (or π/6 radians). They are used to find an angle when you know the ratio of two sides. Note that inverse functions have restricted ranges to ensure a unique output: arcsin returns [−90°, 90°], arccos returns [0°, 180°], and arctan returns (−90°, 90°).

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. For any angle θ measured counterclockwise from the positive x-axis, the point on the unit circle is (cos θ, sin θ). This geometric interpretation makes it easy to determine exact values of trig functions at standard angles (0°, 30°, 45°, 60°, 90°, etc.) and to understand why sine and cosine are periodic with period 2π.