The **Trigonometric Calculator** is an essential tool for solving unknown sides or angles of a right-angled triangle using the fundamental SOH CAH TOA principles. Input any two values (excluding the right angle) to solve the complete triangle.
Trigonometric Calculator
Trigonometric Calculator Formula:
CAH: $\cos(\text{Angle}) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
TOA: $\tan(\text{Angle}) = \frac{\text{Opposite}}{\text{Adjacent}}$
Pythagorean Theorem: $a^2 + b^2 = c^2$
Angle Sum: $A + B + 90^\circ = 180^\circ$
Formula Sources: Math Is Fun, BYJU’S Education
Variables:
- Side ‘a’ (Opposite): The length of the side opposite Angle A.
- Side ‘b’ (Adjacent): The length of the side adjacent (next to) Angle A.
- Hypotenuse ‘c’: The longest side, opposite the $90^\circ$ right angle.
- Angle ‘A’ (Degrees): One of the two acute angles (non-90 degree) in the triangle.
- Angle ‘B’ (Degrees): The other acute angle, calculated as $90^\circ – A$.
Related Calculators:
What is a Trigonometric Calculator?
A Trigonometric Calculator is a specialized mathematical tool designed to solve for unknown elements—sides or angles—within a triangle, typically a right-angled triangle. It operates based on trigonometric ratios (Sine, Cosine, and Tangent) that describe the relationship between the angles and side lengths of a triangle.
Trigonometry, derived from Greek words meaning “triangle measure,” is fundamental in fields like engineering, physics, surveying, navigation, and astronomy. This calculator automates complex calculations, allowing users to find missing values quickly and accurately, which is essential for applied geometry problems.
How to Calculate Trigonometric Values (Example):
Let’s find the missing side ‘a’ and angles A and B, given Side ‘b’ = 8 and Hypotenuse ‘c’ = 10:
- Find Side ‘a’ (Opposite) using Pythagoras: The formula is $a^2 + b^2 = c^2$. Substitute known values: $a^2 + 8^2 = 10^2$, so $a^2 + 64 = 100$. This gives $a^2 = 36$, meaning Side ‘a’ = 6.
- Find Angle ‘A’ using SOH (Sine): $\sin(A) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{c} = \frac{6}{10} = 0.6$. The angle is found using the inverse sine function: $A = \arcsin(0.6)$, which is approximately $36.87^\circ$.
- Find Angle ‘B’ using Angle Sum: Since the angles sum to $180^\circ$ and the right angle is $90^\circ$, $B = 90^\circ – A$. $B = 90^\circ – 36.87^\circ = 53.13^\circ$.
- Verify: All sides (6, 8, 10) and angles ($36.87^\circ, 53.13^\circ, 90^\circ$) are now known.
Frequently Asked Questions (FAQ):
What is the difference between $\sin$, $\cos$, and $\tan$?
They are ratios relating the sides of a right triangle to its acute angles. Sine ($\sin$) is the ratio of the opposite side to the hypotenuse. Cosine ($\cos$) is the ratio of the adjacent side to the hypotenuse. Tangent ($\tan$) is the ratio of the opposite side to the adjacent side.
Can this calculator solve for non-right triangles?
No. This specific calculator is designed only for right-angled triangles ($90^\circ$ angle). Solving non-right triangles requires the Law of Sines or the Law of Cosines, which are different formulas.
What units do the side lengths need to be in?
The side lengths can be in any unit (meters, feet, inches) as long as you are consistent. The resulting calculated side lengths will be in the same unit. Angles must always be input and calculated in degrees.
Why did I get an error message?
Errors usually occur if you provide only one input, or if the side lengths violate the Pythagorean theorem (e.g., the hypotenuse ‘c’ is shorter than one of the sides ‘a’ or ‘b’), which is geometrically impossible.