Trigonometric Functions - Unit-circle Definitions

Unit-circle Definitions

The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles.

The unit circle definition does, however, permit the definition of the trigonometric functions for all positive and negative arguments, not just for angles between 0 and π/2 radians.

It also provides a single visual picture that encapsulates at once all the important triangles. From the Pythagorean theorem the equation for the unit circle is:

In the picture, some common angles, measured in radians, are given. Measurements in the counterclockwise direction are positive angles and measurements in the clockwise direction are negative angles.

Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively.

The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we have sin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.

Note that these values can easily be memorized in the form

but the angles are not equally spaced.

The values for 15°, 18º, 36º, 54°, 72º, and 75° are derived as follows.

the values for 3º, 6º, 9º, 81º, 84º, and 87º can also be computed analytically.

For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine and cosine are periodic functions with period 2π:

for any angle θ and any integer k.

The smallest positive period of a periodic function is called the primitive period of the function.

The primitive period of the sine or cosine is a full circle, i.e. 2π radians or 360 degrees.

Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be defined by:

So :

• The primitive period of the secant, or cosecant is also a full circle, i.e. 2π radians or 360 degrees.
• The primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.

The image at right includes a graph of the tangent function.

• Its θ-intercepts correspond to those of sin(θ) while its undefined values correspond to the θ-intercepts of cos(θ).
• The function changes slowly around angles of kπ, but changes rapidly at angles close to (k + 1/2)π.
• The graph of the tangent function also has a vertical asymptote at θ = (k + 1/2)π, the θ-intercepts of the cosine function, because the function approaches infinity as θ approaches (k + 1/2)π from the left and minus infinity as it approaches (k + 1/2)π from the right.

Alternatively, all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (as shown in the picture to the right), and similar such geometric definitions were used historically.

• In particular, for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord), a definition introduced in India (see history).
• cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD.
• tan(θ) is the length of the segment AE of the tangent line through A, hence the word tangent for this function. cot(θ) is another tangent segment, AF.
• sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively.
• DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle).
• From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions are possible, and the basic trigonometric identities can also be proven graphically.)