Trigonometry Calculator
Calculate trigonometric functions including sine, cosine, tangent, and their inverses with step-by-step solutions.
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Trigonometry Calculator
Solve trigonometric problems with detailed step-by-step solutions. Select a trigonometric function and enter the required values.
How to Input Values
- For sine (sin), cosine (cos), tangent (tan): Enter the angle in degrees, or input the side lengths (opposite, adjacent, hypotenuse) to calculate or verify the result.
- For arcsine (sin⁻¹): Enter opposite side and hypotenuse lengths (required) to find the angle. Angle and adjacent side are optional and disabled.
- For arccosine (cos⁻¹): Enter adjacent side and hypotenuse lengths (required) to find the angle. Angle and opposite side are optional and disabled.
- For arctangent (tan⁻¹): Enter opposite and adjacent side lengths (required) to find the angle. Angle and hypotenuse are optional and disabled.
- Note: Ensure side lengths are positive. For inverse functions, ratios must be valid (e.g., opposite ≤ hypotenuse).
These results are for reference only and were developed for educational and testing purposes. You can also directly access and review the source code, including the logic and free APIs used on this page.
Calculation Results
Trigonometry Calculator Guide
The Trigonometry Calculator is a tool designed to calculate trigonometric functions and their inverses with detailed step-by-step solutions. This guide provides instructions for using the calculator and objective information about trigonometric functions.
How to Use the Trigonometry Calculator
Follow these steps to calculate trigonometric functions:
- Select Function: Choose a trigonometric function (sine, cosine, tangent, or their inverses) from the dropdown menu.
- Enter Values: Input the angle (in degrees) or side lengths (opposite, adjacent, hypotenuse) as required. Optional fields are marked, and unnecessary fields are disabled.
- Calculate: Click "Calculate" to view the result and detailed solution steps.
- Review Results: Examine the calculated result, formula used, and step-by-step solution.
About Trigonometric Functions
Trigonometric functions describe relationships between angles and sides in a right-angled triangle. They are fundamental in geometry, physics, and engineering.
Basic Formulas
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Right Triangle Diagram
This diagram illustrates a right triangle where:
- Opposite: The side opposite the angle θ.
- Adjacent: The side next to θ (not the hypotenuse).
- Hypotenuse: The longest side, opposite the right angle.
Graphical Representation
Trigonometric functions are periodic and can be visualized as waves. Below are graphs of sine and cosine functions over one period (0° to 360°):
Sine and Cosine Graphs
The sine function (blue) and cosine function (red) oscillate between -1 and 1, with a period of 360°. The tangent function is different:
Tangent Graph
The tangent function (purple) has vertical asymptotes at 90° and 270°, where it is undefined, and repeats every 180°.
Inverse Trigonometric Functions
Inverse functions return angles based on side ratios:
\[ \theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right) \]
\[ \theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right) \]
\[ \theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) \]
- Arcsine (sin⁻¹): Returns angles between -90° and 90°.
- Arccosine (cos⁻¹): Returns angles between 0° and 180°.
- Arctangent (tan⁻¹): Returns angles between -90° and 90°.
Mathematical Concepts
- Relate angles of a triangle to the lengths of its sides.
- Fundamental in geometry, physics, and engineering.
- Periodic functions with applications in wave analysis.
- Inverse functions return angles from ratios.
Final Tips for Using the Calculator
- Ensure you select the correct trigonometric function for your problem.
- Enter positive values for side lengths.
- For inverse functions, ensure ratios are valid (e.g., opposite ≤ hypotenuse).
- Review the step-by-step solution to understand the calculation process.
- Angles are in degrees; convert if working in radians.
The calculator provides solutions based on standard trigonometric formulas. For specialized applications, consult additional references.