sin(45°)
0.707107
cos
0.7071
tan
1.0000
0.707107
0.707107
1
1.414214
1.414214
1
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.8660 | 0.5774 |
| 45° | 0.7071 | 0.7071 | 1 |
| 60° | 0.8660 | 0.5 | 1.7321 |
| 90° | 1 | 0 | undef |
sin(θ) = opp/hyp, cos(θ) = adj/hyp, tan(θ) = opp/adjThe three primary trigonometric ratios relate the sides of a right triangle to its angles. These are the building blocks for all trigonometric calculations.
Where:
θ= The angle being evaluatedopp= Side opposite to the angleadj= Side adjacent to the anglehyp= Hypotenuse (longest side, opposite 90°)radians = degrees × π / 180Converts an angle from degree measure to radian measure. One full revolution equals 360 degrees or 2π radians.
Where:
degrees= Angle in degrees (0° to 360°)π= Pi, approximately 3.14159radians= Angle in radianssin²(θ) + cos²(θ) = 1The fundamental trigonometric identity. For any angle, the square of sine plus the square of cosine always equals 1. Derived from the unit circle and the Pythagorean theorem.
Where:
θ= Any angle in degrees or radianssin²= Sine value squaredcos²= Cosine value squaredInputs
Result
sin(30°) = 0.5 is one of the exact values from the unit circle. In a 30-60-90 triangle, the side opposite 30° is exactly half the hypotenuse.
Inputs
Result
cos(60°) = 0.5 is a complementary identity: cos(60°) = sin(30°). In a 30-60-90 triangle, the adjacent side to 60° is half the hypotenuse.
Inputs
Result
tan(45°) = 1 because sin(45°) = cos(45°). In a 45-45-90 isosceles right triangle, the two legs are equal, making their ratio exactly 1.
The 6 trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). The first three are primary functions; the last three are their reciprocals.
| Function | At 30° | At 45° | At 60° |
|---|---|---|---|
| sin | 0.5 | 0.7071 | 0.8660 |
| cos | 0.8660 | 0.7071 | 0.5 |
| tan | 0.5774 | 1 | 1.7321 |
| csc | 2 | 1.4142 | 1.1547 |
SOH-CAH-TOA is a mnemonic for remembering the three primary trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It works for any right triangle.
| Mnemonic | Formula | Example (3-4-5) |
|---|---|---|
| SOH | sin = opp/hyp | sin(A) = 3/5 = 0.6 |
| CAH | cos = adj/hyp | cos(A) = 4/5 = 0.8 |
| TOA | tan = opp/adj | tan(A) = 3/4 = 0.75 |
Degrees and radians are two units for measuring angles. A full circle is 360 degrees or 2π radians. To convert: radians = degrees × π/180. Radians are preferred in calculus and physics because they simplify many formulas.
| Degrees | Radians | Fraction of π |
|---|---|---|
| 0° | 0 | 0 |
| 30° | 0.5236 | π/6 |
| 45° | 0.7854 | π/4 |
| 90° | 1.5708 | π/2 |
Inverse trig functions (arcsin, arccos, arctan) work backwards from a trig value to find the angle. For example, if sin(θ) = 0.5, then arcsin(0.5) = 30°. Arcsin and arccos accept inputs from -1 to 1; arctan accepts any real number.
| Function | Input | Output (degrees) |
|---|---|---|
| arcsin | 0.5 | 30° |
| arccos | 0.5 | 60° |
| arctan | 1 | 45° |
| arctan | 0 | 0° |
The unit circle is a circle with radius 1 centered at the origin. Any point (x, y) on it satisfies x = cos(θ) and y = sin(θ). It provides exact values for all standard angles and is the foundation of trigonometry.
| Angle | cos (x) | sin (y) |
|---|---|---|
| 0° | 1 | 0 |
| 90° | 0 | 1 |
| 180° | -1 | 0 |
| 270° | 0 | -1 |
sin(30°) = 0.5 exactly, cos(60°) = 0.5 exactly, and tan(45°) = 1 exactly — these values come from the special right triangles (30-60-90 and 45-45-90) and the unit circle. The three primary functions (sine, cosine, tangent) relate sides of a right triangle to its angles: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
The three reciprocal functions are csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = 1/tan(θ). At 30°: csc = 2, sec = 1.1547, cot = 1.7321. At 90°, sin = 1 and cos = 0, making tan and sec undefined (division by zero). These undefined points create vertical asymptotes in the graphs of tan and sec.
The mnemonic SOH-CAH-TOA encodes the three primary ratios. For a 3-4-5 right triangle with angle A opposite the side of length 3: sin(A) = 3/5 = 0.6, cos(A) = 4/5 = 0.8, tan(A) = 3/4 = 0.75. The angle A = arcsin(0.6) = 36.87°. The Pythagorean Theorem Calculator verifies the side relationships while this calculator handles the angle computations.
| Angle | sin | cos | tan | Radians |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | 0.5 | 0.8660 | 0.5774 | π/6 |
| 45° | 0.7071 | 0.7071 | 1 | π/4 |
| 60° | 0.8660 | 0.5 | 1.7321 | π/3 |
| 90° | 1 | 0 | undefined | π/2 |
A full circle is 360° or 2π radians. The conversion factor is π/180 ≈ 0.017453: to convert degrees to radians, multiply by π/180. A right angle (90°) = π/2 ≈ 1.5708 radians. One radian = 180°/π ≈ 57.296°, which is the angle subtended by an arc equal in length to the radius.
Radians are preferred in calculus because they simplify derivative formulas: d/dx[sin(x)] = cos(x) only when x is in radians. In degrees, d/dx[sin(x°)] = (π/180)·cos(x°), adding an extra factor. Physics and engineering formulas (angular velocity ω = Δθ/Δt, arc length s = rθ) also assume radians.
Common conversions: 45° = π/4 ≈ 0.7854 rad, 120° = 2π/3 ≈ 2.0944 rad, 270° = 3π/2 ≈ 4.7124 rad. Negative angles measure clockwise rotation: −90° = −π/2. Angles beyond 360° wrap around: 450° = 360° + 90°, same position as 90°. The Circle Calculator uses radians for arc length and sector area formulas.
Inverse trig functions recover angles from side ratios. arcsin(0.5) = 30° because sin(30°) = 0.5. arccos(0) = 90° because cos(90°) = 0. arctan(1) = 45° because tan(45°) = 1. These functions are essential when you know side lengths and need the angle.
Each inverse function has a restricted range to ensure a unique output. arcsin: [−90°, 90°], arccos: [0°, 180°], arctan: (−90°, 90°). For angles outside these ranges, use reference angles and quadrant analysis. For example, sin(150°) = 0.5, but arcsin(0.5) returns 30°, not 150°. You need to recognize that 150° is in Quadrant II where sine is positive and adjust accordingly.
Real-world applications include surveying (measuring angles from distance ratios), navigation (bearing calculations from coordinate offsets), and physics (resolving force vectors into components). A ladder leaning against a wall at 12 feet high with the base 5 feet from the wall makes angle arctan(12/5) = 67.38° with the ground. The Right Triangle Calculator automates these angle-from-sides calculations with full step-by-step solutions.
The atan2(y, x) function (available in most programming languages) handles all four quadrants automatically by considering the signs of both y and x. Standard arctan(y/x) cannot distinguish between Quadrants I and III, or II and IV.
The Pythagorean identity sin²(θ) + cos²(θ) = 1 holds for every angle. At 30°: 0.25 + 0.75 = 1. At 45°: 0.5 + 0.5 = 1. Dividing through by cos² gives 1 + tan²(θ) = sec²(θ), and dividing by sin² gives cot²(θ) + 1 = csc²(θ). These three Pythagorean identities are used constantly in simplification and proof problems.
Double angle formulas: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) − sin²(θ). At θ = 30°: sin(60°) = 2(0.5)(0.866) = 0.866. Sum formulas: sin(A+B) = sinA·cosB + cosA·sinB. These identities are essential for signal processing, where combining sine waves of different frequencies requires expanding products of trigonometric functions.
In calculus, the identity sin(x)/x → 1 as x → 0 (x in radians) is the foundation for differentiating trigonometric functions. The derivative d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = −sin(x) create a cycle that repeats every 4 derivatives. These patterns make trig functions natural solutions to differential equations modeling oscillation, waves, and alternating current circuits.
Last Updated: Mar 26, 2026
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