What is the SOH CAH TOA rule in trigonometry?

SOH CAH TOA is a mnemonic for right-triangle trigonometry: Sine = Opposite / Hypotenuse (SOH), Cosine = Adjacent / Hypotenuse (CAH), and Tangent = Opposite / Adjacent (TOA).

How does the Law of Sines work for ambiguous SSA cases?

When given two sides and a non-included acute angle (SSA), comparing the opposite side to the triangle altitude h = b*sin(A) determines if there are 0 solutions (side too short), exactly 1 right or obtuse solution, or 2 distinct valid triangles.

How do you convert degrees to radians?

To convert an angle from degrees to radians, multiply the degree value by π / 180. For example, 180° = π radians, and 90° = π/2 radians.

What are the six geometric lines on the unit circle?

On the unit circle, Sine is the vertical coordinate, Cosine is the horizontal coordinate, Tangent is the tangent line segment at x=1, Secant is the ray to the tangent line, Cotangent is the horizontal tangent segment at y=1, and Cosecant is the ray to the cotangent line.

Trigonometric Calculator | Sin, Cos, Tan, Triangles & Unit Circle

Trigonometric Function Solver

Calculate 6 standard & reciprocal functions with exact radicals

Mode: Degrees (°)

Special Angles:

Angle Value (θ) Accepts numbers or fractions like pi/4

θ =

Primary Function

Function Category:

TrigNova Radial Angle Compass

sin(45°) Quadrant I

0.707107

Exact: √2 / 2

Angle Geometry & Step-by-Step Breakdown

Angle in Radians & Degrees:

θ = 45° = π/4 rad ≈ 0.785398 rad

Reference Angle (θ_ref):

θ_ref = 45° (In Quadrant I, θ_ref = θ)

ASTC Sign Rule:

Quadrant I ("All"): All trigonometric values are positive (+)

Right Triangle Solver (90°)

Geometric constructions with Incircle, Circumcircle, and Altitude

Classic Triangles:

Side a (Adjacent to B, Opp to A) Base leg

a =

Side b (Adjacent to A, Opp to B) Height leg

b =

Hypotenuse c (Opposite 90° Angle C) Longest side

c =

Angle α (A) Opposite side a

α = °

Angle β (B) Opposite side b

β = °

TrigNova Geometric Construction

Side a (Base) 3.0000

Side b (Height) 4.0000

Hypotenuse c 5.0000

Angle α (A) 36.8699°

Angle β (B) 53.1301°

Altitude (h_c) 2.4000

Inradius (r) 1.0000

Circumradius (R) 2.5000

Area (1/2·a·b) 6.0000

Pythagorean & Geometric Derivations

Pythagorean Theorem: \(c = \sqrt{a^2 + b^2}\)

\(c = \sqrt{3^2 + 4^2} = 5.0000\)

Finding Angle α using Inverse Tangent: \(\tan(\alpha) = a / b\)

\(\alpha = \arctan(3 / 4) \approx 36.8699^\circ\)

Altitude to Hypotenuse: \(h_c = (a \cdot b) / c\)

\(h_c = (3 \cdot 4) / 5 = 2.4000\)

Oblique Triangle Solver

Solve any triangle via Law of Sines & Cosines (SSS, SAS, ASA, AAS, SSA)

Select Given Information (Method)

Side a

a =

Side b

b =

Side c

c =

TrigNova Altitude & Law of Sines Visualizer

Side a 7.0000

Side b 8.0000

Side c 9.0000

Angle α (A) 48.1897°

Angle β (B) 58.4119°

Angle γ (C) 73.3984°

Altitude (h) 5.9628

Circumradius (R) 4.6953

Area (Heron's) 26.8328

Step-by-Step Law of Cosines / Sines Solution

Unit Circle & 6 Trig Lines

Geometric definition: sin, cos, tan, cot, sec, and csc line segments

Angle Slider (θ) 45° (π/4 rad)

sin θ (Drop leg)

cos θ (Base leg)

tan θ (Tangent x=1)

sec θ (Origin ray)

cot θ (Tangent y=1)

csc θ (Origin ray)

Snap to Unit Circle Angles:

0°

30°

π/6

45°

π/4

60°

π/3

90°

π/2

120°

2π/3

135°

3π/4

150°

5π/6

180°

210°

7π/6

225°

5π/4

240°

4π/3

270°

3π/2

300°

5π/3

315°

7π/4

330°

11π/6

ASTC Signs Rule: "All Students Take Calculus"

A
Quad I (0°-90°)
ALL +

Quad II (90°-180°)

SIN, CSC +

Quad III (180°-270°)

TAN, COT +

Quad IV (270°-360°)

COS, SEC +

Interactive 6 Geometric Trig Lines

Terminal Point Coordinates (x, y) Quadrant I

(0.7071, 0.7071)

(cos θ, sin θ) = (√2/2, √2/2)

cos(θ) = x 0.7071

sin(θ) = y 0.7071

tan(θ) = y/x 1.0000

sec(θ) = 1/x 1.4142

cot(θ) = x/y 1.0000

Reference Angle 45.00°

Rolling Wave Generator & Grapher

Visual connection: rotating circle projecting onto continuous wave

Waveform Function

Amplitude (A) 1.00

Frequency / Period factor (B) 1.00

Phase Shift (C) 0.00 rad

Vertical Shift (D) 0.00

TrigNova Synchronized Rolling Circle Tracer

Period (T = 2π/B) 6.2832 rad

Frequency (f = 1/T) 0.1592 Hz

Max Crest (D + A) 1.0000

Min Trough (D - A) -1.0000

Live Multi-Unit Angle Converter

Instant conversions across 6 angle units with exact π-fraction notation

Input Angle Magnitude

Degrees (°) 180.0000°

Radians (rad) 3.14159 rad Exact: π rad

Gradians (gon) 200.0000 gon

Turns (Revolutions) 0.5000 rev

Arcminutes (′) 10,800.00′

Arcseconds (″) 648,000.00″

Pythagorean Identities Core

\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta

The three fundamental relations derived from the unit circle equation \(x^2 + y^2 = 1\).

Reciprocal & Quotient Fundamentals

\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$$ $$\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}

Defines cosecant, secant, and cotangent in terms of sine and cosine.

Sum & Difference Formulas Compound

\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$ $$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$ $$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}

Crucial for expanding composite angles and calculating exact non-standard angles.

Double-Angle Formulas Calculus Core

\sin(2\theta) = 2\sin\theta\cos\theta$$ $$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1$$ $$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

Expresses trigonometric functions of twice an angle in terms of single-angle functions.

Half-Angle Formulas Radicals

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$ $$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$ $$\tan\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}

Signs \(\pm\) depend on the quadrant in which \(\theta/2\) lies.

Law of Sines & Cosines Triangles

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$ $$c^2 = a^2 + b^2 - 2ab\cos C

Enables solving any acute or obtuse triangle where \(R\) is the circumradius.

Product-to-Sum Formulas Integration

\sin A\cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$$ $$\cos A\cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$$ $$\sin A\sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]

Indispensable in Fourier analysis, audio synthesis, and calculus integration.

Even & Odd Symmetries Symmetry

\sin(-\theta) = -\sin\theta \quad (\text{Odd})$$ $$\cos(-\theta) = \cos\theta \quad (\text{Even})$$ $$\tan(-\theta) = -\tan\theta \quad (\text{Odd})

Cosine is symmetrical across the y-axis, while sine and tangent are rotational odd functions.

Applied Word Problem Solvers

Real-world scenarios with dynamic geometric illustrations

Select Scenario

Distance to Object (d)

d = m

Angle of Elevation (θ)

θ = °

Observer Eye Level Height (h0)

h0 = m

TrigNova Scenario Pictorial Illustration

Calculated Building Height (H)

36.71 m

H = d · tan(θ) + h0

Detailed Solution Steps

Trigonometry Mastery Challenge

Question 1 of 5

Score: 0 / 0

What is the exact value of \(\sin(30^\circ)\)?

High-Precision Trigonometric Calculator & Geometric Suite