Enter any 2 (sides and angles) values to calculate the remaining trigonometric ratios using the SOHCATOA mnemonic.
The calculator helps to solve right-angled triangles using trigonometric ratios based on mnemonic SOHCAHTOA. It provides you with accurate values for each side and angle of a right triangle and shows complete steps for better understanding.
The calculator requires only 2 inputs to calculate results:
Input 1 | Input 2 | Valid? | Reason |
---|---|---|---|
Side a | Side b | ✓ | Use Pythagorean theorem → find hypotenuse and angles |
Side a | Side c | ✓ | Use Pythagoras and trig ratios → find b and angles |
Side b | Side c | ✓ | Same as above |
Side a | Angle α | ✓ | Use SOH (sin α = a/c) → find c, then b and β |
Side b | Angle α | ✓ | Use TOA (tan α = a/b) or CAH |
Side c | Angle α | ✓ | Use SOH and CAH |
Side a | Angle β | ✓ | Similar process, using sin β |
Side b | Angle β | ✓ | Use tan β = b/a |
Side c | Angle β | ✓ | Use sin β = b/c |
Input 1 | Input 2 | Valid? | Reason |
---|---|---|---|
Angle α | Angle β | ✗ | Only gives full angle info; no side lengths can be determined |
Angle α | None | ✗ | Angle alone doesn’t define triangle size |
Angle β | None | ✗ | Same as above |
Side a | None | ✗ | Only one side - can’t determine angles or other sides |
Side b | None | ✗ | Same issue |
Side c | None | ✗ | Need one more piece of information |
Features:
It’s a mnemonic device that anyone can use to remember the three basic trigonometric ratios to determine the missing sides and angles of the right-angled triangle.
It is the combination of three functions: SOH, CAH, and TOA.
The mnemonic sohcahtoa is used to remember which function is used in what circumstances and is also used to find the trigonometric ratios of an acute angle of a triangle.
sin(θ) | cos(θ) | tan(θ) = sin(θ) / cos(θ) | |
---|---|---|---|
0° = 0 radians | √0/2 = 0 | √4/2 = 1 | 0 / 1 = 0 |
30° = π/6 radians | √1/2 = 1/2 | √3/2 | (1/2) / (√3/2) = 1/√3 |
45° = π/4 radians | √2/2 = 1/√2 | √2/2 = 1/√2 | (1/√2) / (1/√2) = 1 |
60° = π/3 radians | √3/2 | √1/2 = 1/2 | (√3/2) / (1/2) = √3 |
90° = π/2 radians | √4/2 = 1 | √0/2 = 0 | 1 / 0 = Undefined |
Given:
a = 4 cm
β = 10°
We are to find the other sides b
, c
, the remaining angle α
, and the area using trigonometric identities and formulas.
Using the sine ratio: sin(β) = a / c
c = a / sin(β) = 4 / sin(10°) ≈ 4 / 0.1736481777
c ≈ 4.0617 cm
Using the Pythagorean Theorem: b = √(c² - a²)
b = √(4.0617² - 4²) = √(16.4975 - 16)
b ≈ √0.4975 ≈ 0.7053 cm
c = √(a² + b²) = √(4² + 0.7053²) = √(16 + 0.4975)
c ≈ √16.4975 ≈ 4.0617 cm
Using tangent ratio: tan(α) = a / b
α = arctan(4 / 0.7053) = arctan(5.6713)
α ≈ 79.0° (or 1.3963 radians)
Area = ½ × a × b = ½ × 4 × 0.7053
Area ≈ 1.4106 cm²
In calculus and analytic geometry, these ratios have a wide range of uses. So which are these and how these are calculated? Read on!
1. Sine:
Sine = Perpendicular / Hypotenuse
2. Cosine:
Cosine = Base / Hypotenuse
3. Tangent:
Tangent = Perpendicular / Base
4. Secant:
Secant = Hypotenuse / Perpendicular
5. Cosecant:
Cosecant = Hypotenuse / Base
6. Cotangent:
Cotangent = Base / Perpendicular
1. Arcsine:
Arcsine = sin^{-1}x
2. Arccosine:
Arccosine = cos^{-1}x
3. Arctangent:
Arctangent = tan^{-1}x
4. Arcsecant:
Arcsecant = sec^{-1}x
5. Arccosecant:
Arccosecant = cosec^{-1}x
6. Arcotangent:
Arcotangent = cot^{-1}x
Another collective phrase commonly used to recall the trig functions is seen below:
“Oscar Had a Heap of Apples”
This implies that:
Not always. You can use the calculator if you know at least one angle (besides the right angle) and one side, or if you know two sides. The SOH CAH TOA calculator will use trigonometric ratios to find the missing sides or angles, depending on what you enter.
By default, our SOHCAHTOA calculator uses both degrees and radians, since these are the most common formats in school problems and real-world use.
In order to find the missing side of the right angle triangle we use the Pythagorean theorem and in order to memorize the trigonometric functions easily we use the sohcahtoa.
From the source Wikipedia: Mnemonics in trigonometry, sohcahtoa how to find angle, Hexagon chart.
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