Right Triangle Angle Calculator

Right Triangle Angle Formula:

\[ \theta = \arcsin\left(\frac{\text{opposite}}{\text{hypotenuse}}\right) \]

Angle (θ):

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1. What is the Right Triangle Angle Formula?

The right triangle angle formula calculates an angle (θ) using the ratio of the opposite side length to the hypotenuse length. This is based on the trigonometric sine function.

2. How Does the Calculator Work?

The calculator uses the inverse sine (arcsine) function:

\[ \theta = \arcsin\left(\frac{\text{opposite}}{\text{hypotenuse}}\right) \]

Where:

\( \theta \) — Angle in degrees or radians
opposite — Length of side opposite to the angle
hypotenuse — Longest side of right triangle

Explanation: The arcsine function returns the angle whose sine equals the ratio of opposite side to hypotenuse.

3. Importance of Angle Calculation

Details: Calculating angles in right triangles is fundamental in trigonometry, used in engineering, physics, architecture, and navigation.

4. Using the Calculator

Tips: Enter positive lengths where the opposite side ≤ hypotenuse. Select desired output unit (degrees or radians).

5. Frequently Asked Questions (FAQ)

Q1: Why must opposite ≤ hypotenuse?
A: In right triangles, the hypotenuse is always the longest side, so the ratio opposite/hypotenuse must be ≤1 for real solutions.

Q2: When should I use degrees vs radians?
A: Degrees are more common in everyday applications, while radians are preferred in higher mathematics and physics.

Q3: What's the range of possible angles?
A: In a right triangle, non-right angles range between 0° and 90° (0 to π/2 radians).

Q4: Can I calculate the other angle?
A: Yes, the other non-right angle is 90° minus the calculated angle (or π/2 minus in radians).

Q5: What if I know adjacent side instead of opposite?
A: Use arccosine(adjacent/hypotenuse) or arctangent(opposite/adjacent) instead.