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Trigonometry

The study of angles and of the angular relationships of planar and three-dimensional figures is known as trigonometry. The trigonometric functions (also called the circular functions) comprising trigonometry are the cosecant cscx, cosine cosx, cotangent cotx, secant secx, sine sinx, and tangent tanx. The inverses of these functions are denoted csc^(-1)x, cos^(-1)x, cot^(-1)x, sec^(-1)x, sin^(-1)x, and tan^(-1)x. Note that the f^(-1) notation here means inverse function, not f to the -1 power.

TrigonometryUnitCircle

The trigonometric functions are most simply defined using the unit circle. Let theta be an angle measured counterclockwise from the x-axis along an arc of the circle. Then costheta is the horizontal coordinate of the arc endpoint, and sintheta is the vertical component. The ratio sintheta/costheta is defined as tantheta. As a result of this definition, the trigonometric functions are periodic with period 2pi, so

func(2pin+theta)=func(theta),
(1)

where n is an integer and func is a trigonometric function.

TrigonometryMnemonic

A right triangle has three sides, which can be uniquely identified as the hypotenuse, adjacent to a given angle theta, or opposite theta. A helpful mnemonic for remembering the definitions of the trigonometric functions is then given by "oh, ah, o-a," "Soh, Cah, Toa," or "SOHCAHTOA", i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent,

sintheta=(opposite)/(hypotenuse)
(2)
costheta=(adjacent)/(hypotenuse)
(3)
tantheta=(opposite)/(adjacent).
(4)

Another mnemonic probably more common in Great Britain than the United States is "Tommy On A Ship Of His Caught A Herring."

From the Pythagorean theorem,

sin^2theta+cos^2theta=1.
(5)

It is therefore also true that

tan^2theta+1=sec^2theta
(6)

and

1+cot^2theta=csc^2theta.
(7)

The trigonometric functions can be defined algebraically in terms of complex exponentials (i.e., using the Euler formula) as

sinz=(e^(iz)-e^(-iz))/(2i)
(8)
cscz=1/(sinz)
(9)
=(2i)/(e^(iz)-e^(-iz))
(10)
cosz=(e^(iz)+e^(-iz))/2
(11)
secz=1/(cosz)
(12)
=2/(e^(iz)+e^(-iz))
(13)
tanz=(sinz)/(cosz)
(14)
=(e^(iz)-e^(-iz))/(i(e^(iz)+e^(-iz)))
(15)
cotz=1/(tanz)
(16)
=(i(e^(iz)+e^(-iz)))/(e^(iz)-e^(-iz))
(17)
=(i(1+e^(-2iz)))/(1-e^(-2iz)).
(18)

Hybrid trigonometric product/sum formulas are

sin(alpha+beta)sin(alpha-beta)=sin^2alpha-sin^2beta
(19)
=cos^2beta-cos^2alpha
(20)
cos(alpha+beta)cos(alpha-beta)=cos^2alpha-sin^2beta
(21)
=cos^2beta-sin^2alpha.
(22)

Osborn's rule gives a prescription for converting trigonometric identities to analogous identities for hyperbolic functions.

For imaginary arguments,

sin(iz)=isinhz
(23)
cos(iz)=coshz.
(24)

For complex arguments,

sin(x+iy)=sinxcoshy+icosxsinhy
(25)
cos(x+iy)=cosxcoshy-isinxsinhy.
(26)

For the absolute square of complex arguments z=x+iy,

|sin(x+iy)|^2=sin^2x+sinh^2y
(27)
|cos(x+iy)|^2=cos^2x+sinh^2y.
(28)

The complex modulus also satisfies the curious identity

|sin(x+iy)|=|sinx+sin(iy)|.
(29)

The only functions satisfying identities of this form,

|f(x+iy)|=|f(x)+f(iy)|
(30)

are f(z)=Az, f(z)=Asin(bz), and f(z)=Asinh(bz) (Robinson 1957).

See also

Cosecant, Cosine, Cotangent, Double-Angle Formulas, Euclidean Number, Half-Angle Formulas, Inverse Cosecant, Inverse Cosine, Inverse Cotangent, Inverse Secant, Inverse Sine, Inverse Tangent, Inverse Trigonometric Functions, Multiple-Angle Formulas, Osborn's Rule, Polygon, Prosthaphaeresis Formulas, Secant, Sine, SOHCAHTOA, Tangent, Trigonometric Addition Formulas, Trigonometry Angles, Trigonometric Functions, Trigonometric Power Formulas, Trigonometric Series Formulas, Unit Circle, Werner Formulas Explore this topic in the MathWorld classroom

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Circular Functions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972.Bahm, L. B. The New Trigonometry on Your Own. Patterson, NJ: Littlefield, Adams & Co., 1964.Beyer, W. H. "Trigonometry." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 134-152, 1987.Borchardt, W. G. and Perrott, A. D. A New Trigonometry for Schools. London: G. Bell, 1930.Dixon, R. "The Story of Sine and Cosine." §4.4 in Mathographics. New York: Dover, pp. 102-106, 1991.Hobson, E. W. A Treatise on Plane Trigonometry. London: Cambridge University Press, 1925.Kells, L. M.; Kern, W. F.; and Bland, J. R. Plane and Spherical Trigonometry. New York: McGraw-Hill, 1940.Maor, E. Trigonometric Delights. Princeton, NJ: Princeton University Press, 1998.Morrill, W. K. Plane Trigonometry, rev. ed. Dubuque, IA: Wm. C. Brown, 1964.Robinson, R. M. "A Curious Mathematical Identity." Amer. Math. Monthly 64, 83-85, 1957.Siddons, A. W. and Hughes, R. T. Trigonometry, Part I. London: Cambridge University Press, 1929a.Siddons, A. W. and Hughes, R. T. Trigonometry, Part II. London: Cambridge University Press, 1929b.Siddons, A. W. and Hughes, R. T. Trigonometry, Part III. London: Cambridge University Press, 1929c.Siddons, A. W. and Hughes, R. T. Trigonometry, Part IV. London: Cambridge University Press, 1929d.Sloane, N. J. A. Sequence A003401/M0505 in "The On-Line Encyclopedia of Integer Sequences."Thompson, J. E. Trigonometry for the Practical Man. Princeton, NJ: Van Nostrand, 1946.Yates, R. C. "Trigonometric Functions." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 225-232, 1952.Weisstein, E. W. "Books about Trigonometry." http://www.ericweisstein.com/encyclopedias/books/Trigonometry.html.Zill, D. G. and Dewar, J. M. Trigonometry, 2nd ed. New York: McGraw-Hill 1990.

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Trigonometry

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Weisstein, Eric W. "Trigonometry." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Trigonometry.html

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