Trigonometric Identities
tan t = | sin tcos t | | cot t = | 1tan t | = | cos tsin t |
sec t = | 1cos t | |csc t = | 1sin 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 tangentand 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 andsecant 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 tDouble 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 theseidentities, 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 andsecants.
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 cosinesin (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 t1 – tan s tan t |tan (s – t) = | tan s – tan t1 + tan s tan t |
tan 2t = | 2 tan t1 – tan2 t |
Half-angle formulas
sin t/2 = ±((1 – cos t) / 2)
cos t/2 = ±((1 + cos t) / 2)
tan t/2 = | sin t1 + cos t | = ...
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