Trigonometric Formula

Trigonometric Formula
Useful trig formulas and some very helpful links for learning trigonometric concepts.

Triangle ABC is any triangle with side lengths a,b,c Law of Cosines Law of Sines Summary of trigonometric identities You have seen quite a few trigonometric identities in the past few pages. It is convenient to have a summary of them for reference. These identities mostly refer to one angle denoted t, but there are a few of them involving two angles, and for those, the other angle is denoted s.. More important identities You don't have to know all the identities off the top of your head. But these you should. Defining relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine. tan t = sin t cos t cot t = 1 tan t = cos t sin t sec t = 1 cos t csc t = 1 sin t The Pythagorean formula for sines and cosines. sin2 t + cos2 t = 1 Identities expressing trig functions in terms of their complements cos t = sin(/2 – t) sin t = cos(/2 – t) cot t = tan(/2 – t) tan t = cot(/2 – t) csc t = sec(/2 – t) sec t = csc(/2 – t) Periodicity of trig functions. Sine, cosine, secant, and cosecant have period 2 while tangent and cotangent have period. sin (t + 2) = sin t cos (t + 2) = cos t tan (t + ) = tan t Identities for negative angles. Sine, tangent, cotangent, and cosecant are odd functions while cosine and secant are even functions. sin –t = –sin t cos –t = cos t tan –t = –tan t Sum formulas for sine and cosine sin (s + t) = sin s cos t + cos s sin t cos (s + t) = cos s cos t – sin s sin t Double angle formulas for sine and cosine sin 2t = 2 sin t cos t cos 2t = cos2 t – sin2 t = 2 cos2 t – 1 = 1 – 2 sin2 t Less important identities You should know that there are these identities, but they are not as important as those mentioned above. They can all be derived from those above, but sometimes it takes a bit of work to do so. The Pythagorean formula for tangents and secants. sec2 t = 1 + tan2 t Identities expressing trig functions in terms of their supplements sin( – t) = sin t cos( – t) = –cos t tan( – t) = –tan t Difference formulas for sine and cosine sin (s – t) = sin s cos t – cos s sin t cos (s – t) = cos s cos t + sin s sin t Sum, difference, and double angle formulas for tangent tan (s + t) = tan s + tan t 1 – tan s tan t tan (s – t) = tan s – tan t 1 + tan s tan t tan 2t = 2 tan t 1 – tan2 t Half-angle formulas sin t/2 = ±((1 – cos t) / 2) cos t/2 = ±((1 + cos t) / 2) tan t/2 = sin t 1 + cos t = 1 – cos t sin t Truly obscure identities These are just here for perversity. Yes, of course, they have some applications, but they're usually narrow applications, and they could just as well be forgotten until, if ever, needed. Product-sum identities sin s + sin t = 2 sin s + t 2 cos s – t 2 sin s – sin t = 2 cos s + t 2 sin s – t 2 cos s + cos t = 2 cos s + t 2 cos s – t 2 cos s – cos t = –2 sin s + t 2 sin s – t 2 Product identities sin s cos t = sin (s + t) + sin (s – t) 2 cos s cos t = cos (s + t) + cos (s – t) 2 sin s sin t = cos (s – t) – cos (s + t) 2 Aside: weirdly enough, these product identities were used before logarithms to perform multiplication. Here's how you could use the second one. If you want to multiply x times y, use a table to look up the angle s whose cosine is x and the angle t whose cosine is y. Look up the cosines of the sum s + t, and the difference s – t. Average those two cosines. You get the product xy! Three table look-ups, and computing a sum, a difference, and an average rather than one multiplication. Tycho Brahe (1546-1601), among others, used this algorithm known as prosthaphaeresis. Triple angle formulas. You can easily reconstruct these from the addition and double angle formulas. sin 3t = 3 sin t – 4 sin3 t cos 3t = 4 cos3 t –3 cos t tan 3t = 3 tan t – tan3t 1 – 3 tan2t More half-angle formulas. (These are used in calculus for a particular kind of substitution in integrals sometimes called the Weierstrass t-substitution.) sin t = 2 tan t/2 1 + tan2 t/2 cos t = 1 – tan2 t/2 1 + tan2 t/2 tan t = 2 tan t/2 1 – tan2 t/2 Select topic Introduction Who should take this course? Applications of trigonometry What is trigonometry? Background on geometry Angle measurement Chords Sines Cosines Tangents and slope The trigonometry of right triangles Trigonometric functions Computing trigonometric functions The trigonometry of oblique triangles The laws of sines and cosines Area of a triangle Summary of trigonometric identities Trigonometric Formulas Basic Relations Laws Law of Sines Law of Cosines Radius of Inscribed Circle Radius of Circumscribed Circle Angle Sum Relations Double Angle Relations Multiple Angle Relations Half Angle Relations Production Relations Sum and Difference Relations Power Relations Adding Signals with the Same Frequency or where: Exponential Relations where: Useful Trigonometric Identities This page is meant to be used as reference listing of useful trigonometric identities. No discussion of the proofs or the consequences of these identities will be give here. Pythagorean Identities Addition Formulas Subtraction formulas Double Angle Formulas Half-Angle Formulas Product Formulas Factoring Formulas The following two formulas are of only limited use: