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Circle Trig Definitions

Circle Trig Definitions

A set of definitions of the six trig functions: sine, cosine, tangent, cosecant, secant, and cotangent.

 

Circle with radius r, point (x,y), angle θ. Definitions: sinθ=y/r, cosθ=x/r, tanθ=y/x, cscθ=r/y, secθ=r/x, cotθ=x/y.

 

 

See also

Unit circle trig definitions, SOHCAHTOA

Key Formula

sin⁡θ=yr,cos⁡θ=xr,tan⁡θ=yx\sin\theta = \frac{y}{r}, \quad \cos\theta = \frac{x}{r}, \quad \tan\theta = \frac{y}{x} csc⁡θ=ry,sec⁡θ=rx,cot⁡θ=xy\csc\theta = \frac{r}{y}, \quad \sec\theta = \frac{r}{x}, \quad \cot\theta = \frac{x}{y}
Where:
  • θ\theta = The angle measured from the positive x-axis to the terminal side
  • xx = The x-coordinate of the point on the circle
  • yy = The y-coordinate of the point on the circle
  • rr = The radius of the circle, where r = √(x² + y²), always positive

Worked Example

Problem: A point on a circle centered at the origin has coordinates (3, 4). Find all six trigonometric functions of the angle θ formed with the positive x-axis.
Step 1: Find the radius r using the distance formula.
r=x2+y2=32+42=9+16=25=5r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
Step 2: Calculate sine and cosine using x, y, and r.
sin⁡θ=yr=45,cos⁡θ=xr=35\sin\theta = \frac{y}{r} = \frac{4}{5}, \quad \cos\theta = \frac{x}{r} = \frac{3}{5}
Step 3: Calculate tangent as y over x.
tan⁡θ=yx=43\tan\theta = \frac{y}{x} = \frac{4}{3}
Step 4: Find the reciprocal functions by flipping each ratio.
csc⁡θ=ry=54,sec⁡θ=rx=53,cot⁡θ=xy=34\csc\theta = \frac{r}{y} = \frac{5}{4}, \quad \sec\theta = \frac{r}{x} = \frac{5}{3}, \quad \cot\theta = \frac{x}{y} = \frac{3}{4}
Answer: sin θ = 4/5, cos θ = 3/5, tan θ = 4/3, csc θ = 5/4, sec θ = 5/3, cot θ = 3/4

Another Example

This example uses a point in Quadrant II where x is negative, showing how the signs of trig functions change depending on which quadrant the angle terminates in.

Problem: A point on a circle centered at the origin has coordinates (−5, 12). Find all six trigonometric functions of the angle θ.
Step 1: Find the radius. Note that x is negative, but r is always positive.
r=(−5)2+122=25+144=169=13r = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
Step 2: Calculate sine and cosine. Since x is negative, cosine will be negative.
sin⁡θ=1213,cos⁡θ=−513=−513\sin\theta = \frac{12}{13}, \quad \cos\theta = \frac{-5}{13} = -\frac{5}{13}
Step 3: Calculate tangent. A positive y divided by a negative x gives a negative result.
tan⁡θ=12−5=−125\tan\theta = \frac{12}{-5} = -\frac{12}{5}
Step 4: Find the reciprocal functions.
csc⁡θ=1312,sec⁡θ=13−5=−135,cot⁡θ=−512=−512\csc\theta = \frac{13}{12}, \quad \sec\theta = \frac{13}{-5} = -\frac{13}{5}, \quad \cot\theta = \frac{-5}{12} = -\frac{5}{12}
Answer: sin θ = 12/13, cos θ = −5/13, tan θ = −12/5, csc θ = 13/12, sec θ = −13/5, cot θ = −5/12

Frequently Asked Questions

What is the difference between circle trig definitions and unit circle trig definitions?
Circle trig definitions use a circle of any radius r, so each function involves dividing by r (e.g., sin θ = y/r). Unit circle trig definitions are a special case where r = 1, which simplifies the formulas to sin θ = y and cos θ = x. The circle trig definitions are more general and work for any point in the coordinate plane.
How do you know which trig functions are positive or negative?
The signs depend on the signs of x and y in the quadrant where the angle's terminal side lies. In Quadrant I both x and y are positive, so all six functions are positive. In Quadrant II, x < 0 and y > 0, so only sine and cosecant are positive. In Quadrant III, both are negative, so only tangent and cotangent are positive. In Quadrant IV, x > 0 and y < 0, so only cosine and secant are positive. The mnemonic 'All Students Take Calculus' helps remember this.
Why is r always positive in the circle trig definitions?
The radius r is calculated as √(x² + y²), which is a distance and therefore always positive. Since r is always positive, the signs of the trig functions are determined entirely by the signs of x and y. This is what allows the definitions to work consistently for angles in all four quadrants.

Circle Trig Definitions vs. Unit Circle Trig Definitions

Circle Trig DefinitionsUnit Circle Trig Definitions
RadiusAny radius r = √(x² + y²)Radius is always 1
Sine formulasin θ = y/rsin θ = y
Cosine formulacos θ = x/rcos θ = x
When to useWhen given any point (x, y) not necessarily on the unit circleWhen working directly on the unit circle
GeneralityGeneral — works for any circle centered at the originSpecial case of the general circle definitions

Why It Matters

Circle trig definitions extend trigonometry beyond right triangles to any angle, including angles greater than 90° and negative angles. You encounter them in precalculus and physics whenever you need to evaluate trig functions for points anywhere in the coordinate plane. They form the foundation for understanding trig function signs by quadrant, reference angles, and eventually polar coordinates.

Common Mistakes

Mistake: Using r as a negative value or forgetting to take the positive square root.
Correction: The radius r = √(x² + y²) is always positive. The signs of the trig functions come from x and y, never from r. Even if both coordinates are negative, r remains positive.
Mistake: Mixing up which coordinate goes in the numerator: putting x in the numerator for sine instead of y.
Correction: Remember: sine uses y (both 'sine' and 'y' relate to the vertical direction), while cosine uses x (the horizontal direction). So sin θ = y/r and cos θ = x/r.

Related Terms