SOHCAHTOA Calculator

SOHCAHTOA is a universally recognized memory aid (mnemonic) used in geometry and trigonometry to recall the definitions of the three primary trigonometric ratios for right-angled triangles: Sine, Cosine, and Tangent.

The acronym breaks down into three distinct formula groups:

In any right triangle, the labels "Opposite" and "Adjacent" are defined strictly relative to the selected reference angle ($\theta$). Changing which acute angle you are analyzing swaps the Opposite and Adjacent labels!

Where the variable symbols represent:

• $\theta$ (Theta): The reference acute angle (measured in degrees or radians, where $0^\circ < \theta < 90^\circ$).
• $O$ (Opposite): The leg length positioned directly across from reference angle $\theta$.
• $A$ (Adjacent): The leg length touching reference angle $\theta$ (excluding the hypotenuse).
• $H$ (Hypotenuse): The longest side length opposite the $90^\circ$ right angle.

Diagram Key: $O$ = opposite leg, $A$ = adjacent leg, $H$ = hypotenuse, $\theta$ = reference angle, $90^\circ$ = fixed right angle.

Because all three trigonometric ratios ($\sin\theta, \cos\theta, \tan\theta$) share side measurements from the exact same right triangle, knowing any two values (such as one side and one acute angle, or any two side lengths) allows you to solve the entire triangle completely.

Correctly identifying the three sides of a right triangle is the single most critical step when applying SOHCAHTOA. Follow these clear definitions:

Hypotenuse

The Hypotenuse is always the longest side of the right triangle. It is located directly across from the $90^\circ$ right angle. The hypotenuse never changes position regardless of which reference angle is chosen.

Opposite Side

The Opposite Side is the leg positioned directly across from your selected reference angle $\theta$. If you draw an arrow straight out from the vertex of angle $\theta$, it points directly at the opposite side.

Adjacent Side

The Adjacent Side is the leg that touches your reference angle $\theta$ to help form its corner, but is not the hypotenuse. "Adjacent" literally means "next to."

The Sine ratio compares the length of the opposite leg to the hypotenuse.

Algebraic Rearrangements:

• To find Opposite: $$\text{Opposite} = \text{Hypotenuse} \times \sin(\theta)$$
• To find Hypotenuse: $$\text{Hypotenuse} = \frac{\text{Opposite}}{\sin(\theta)}$$

Worked Example:

Given a right triangle with reference angle $\theta = 30^\circ$ and hypotenuse $H = 10\text{ cm}$. Find the opposite side $O$.

The Cosine ratio compares the length of the adjacent leg to the hypotenuse.

Algebraic Rearrangements:

• To find Adjacent: $$\text{Adjacent} = \text{Hypotenuse} \times \cos(\theta)$$
• To find Hypotenuse: $$\text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)}$$

Worked Example:

Given a right triangle with reference angle $\theta = 60^\circ$ and hypotenuse $H = 20\text{ cm}$. Find the adjacent side $A$.

The Tangent ratio compares the length of the opposite leg directly to the adjacent leg. Tangent does not involve the hypotenuse.

Algebraic Rearrangements:

• To find Opposite: $$\text{Opposite} = \text{Adjacent} \times \tan(\theta)$$
• To find Adjacent: $$\text{Adjacent} = \frac{\text{Opposite}}{\tan(\theta)}$$

Worked Example:

Given a right triangle with reference angle $\theta = 45^\circ$ and adjacent leg $A = 8\text{ m}$. Find the opposite leg $O$.

Find the Opposite Side

If Hypotenuse $H = 12\text{ m}$ and $\theta = 35^\circ$:

Find the Adjacent Side

If Hypotenuse $H = 15\text{ cm}$ and $\theta = 50^\circ$:

Find the Hypotenuse

If Opposite $O = 9\text{ in}$ and $\theta = 28^\circ$:

💡 Understanding $\sin^{-1}$ Notation

In trigonometry, $\sin^{-1}(x)$ means "the angle whose sine is $x$" (inverse sine), NOT $\frac{1}{\sin(x)}$ (which is cosecant, $\csc(x)$). Scientific programming environments use `Math.asin()`, `Math.acos()`, and `Math.atan()`.

Always determine your reference angle $\theta$ first before selecting the ratio!

SOHCAHTOA is built exclusively upon the unique geometric properties of right triangles ($90^\circ$ triangles). In every right triangle:

• The interior angles always sum to $180^\circ$. Because the right angle uses $90^\circ$, the two acute angles are complementary ($\theta + \beta = 90^\circ$).
• The side opposite the $90^\circ$ angle is guaranteed to be the longest side (Hypotenuse).
• The ratios between the side lengths depend strictly on the acute angles, creating constant trigonometric ratios.

Need to solve a general right triangle with both side and angle inputs? Visit our core Right Triangle Calculator.

For pure side-length calculations without angle requirements, use our dedicated Pythagorean Theorem Calculator.

Special right triangles have constant interior angle measurements that produce exact, simple radical SOHCAHTOA ratios:

45-45-90 Isosceles Right Triangle

Side length ratio is $1 : 1 : \sqrt{2}$.

• $\sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.7071$
• $\cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.7071$
• $\tan(45^\circ) = \frac{1}{1} = 1.0$

Explore our specialized 45-45-90 Triangle Calculator.

30-60-90 Right Triangle

Side length ratio is $1 : \sqrt{3} : 2$.

• $\sin(30^\circ) = \frac{1}{2} = 0.5$, $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660$, $\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \approx 0.5774$
• $\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660$, $\cos(60^\circ) = \frac{1}{2} = 0.5$, $\tan(60^\circ) = \sqrt{3} \approx 1.7321$

Explore our specialized 30-60-90 Triangle Calculator.

Pythagorean triples are sets of three whole integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Evaluating SOHCAHTOA for integer triples yields exact rational fractions:

3-4-5 Right Triangle

For the reference angle opposite leg 3:

• $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5} = 0.6$
• $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5} = 0.8$
• $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4} = 0.75$
• Angle $\theta = \sin^{-1}(0.6) \approx 36.87^\circ$.

Other famous integer triples include (5, 12, 13), (8, 15, 17), and (7, 24, 25).

A classic trigonometry problem asks: "A $10\text{ ft}$ ladder rests against a wall, making a $60^\circ$ angle with the ground. How high up the wall does the ladder reach?"

The ladder reaches approximately 8.66 ft up the wall.

Real-world trigonometry problems often involve angles measured from horizontal line-of-sight reference axes:

Angle of Elevation

The angle measured upward from a horizontal reference line to an elevated object (e.g., looking up at the top of a lighthouse).

Angle of Depression

The angle measured downward from a horizontal reference line to an object below (e.g., an observer at the top of a cliff looking down at a boat).

Because alternate interior angles are equal across parallel horizontal lines, the angle of elevation from point A to point B equals the angle of depression from point B to point A!

Our calculator supports seven linear length units: millimeters (mm), centimeters (cm), meters (m), kilometers (km), inches (in), feet (ft), and yards (yd).

Because SOHCAHTOA ratios compare relative side lengths ($\frac{\text{length}}{\text{length}}$), unit dimensions cancel out in the trigonometric ratio! However, all input side lengths must be expressed in the same unit before computing.

Angles can be measured in two common units:

• Degrees (°): Full circle = $360^\circ$, Right angle = $90^\circ$. Standard for geometry and practical measurements.
• Radians (rad): Full circle = $2\pi\text{ rad}$, Right angle = $\frac{\pi}{2}\text{ rad}$. Standard in calculus and advanced mathematics.

Conversion formulas:

Our calculator allows seamless toggling between degree and radian modes.

Trigonometric calculations produce two representation forms:

• Exact Radical Expressions: Values expressed with exact square roots (e.g. $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ or $c = 5\sqrt{2}$).
• Decimal Approximations: Floating-point numbers rounded to specific decimal places (e.g. $\sin(60^\circ) \approx 0.866025$).

Our solver displays exact simplified radical tags alongside 6-decimal rounded values.

Basic SOHCAHTOA ratios apply only to right-angled triangles ($90^\circ$ triangles).

If a triangle does not contain a $90^\circ$ angle (an oblique triangle), SOHCAHTOA cannot be used directly. Instead, you must apply oblique triangle formulas:

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$

SOHCAHTOA Summary Box

• SOH: $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \implies \quad O = H \sin\theta, \quad H = \frac{O}{\sin\theta}$
• CAH: $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \implies \quad A = H \cos\theta, \quad H = \frac{A}{\cos\theta}$
• TOA: $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \quad \implies \quad O = A \tan\theta, \quad A = \frac{O}{\tan\theta}$
• Inverse Functions: $\theta = \sin^{-1}(O/H) = \cos^{-1}(A/H) = \tan^{-1}(O/A)$