Thread: Trigonometric Identities

My teacher was sick the day we learned the trig identities in class and I can't seem to find them in my text book - can someone post the oh say ten or so basic ones I should for sure know? Double angle functions, pythag ones, etc.?

Originally Posted by Pajkaj

My teacher was sick the day we learned the trig identities in class and I can't seem to find them in my text book - can someone post the oh say ten or so basic ones I should for sure know? Double angle functions, pythag ones, etc.?

sin^2(x) + cos^2(x) = 1
tan(x) = sin(x)/cos(x)
sec^2(x) = 1 + tan(x)
csc^2(x) = 1 + cot^2(x)

sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
cot(x) = 1/tan(x) = cos(x)/sin(x)


Addtion and subtraction formulas:

sin(A + B) = sinAcosB + sinBcosA
sin(A - B) = sinAcosB - sinBcosA

cos(A + B) = cosAcosB - sinAsinB
cos(A - B) = cosAcosB + sinAsinB

tan(A + B) = (tanA + tanB)/(1 - tanAtanB)
tan(A - B) = (tanA - tanB)/(1 + tanAtanB)
the tan formulas are not as important to memorize, you can use the identity tan(x) = sin(x)/cos(x) to find them if necessary

Double angle formulas:
for these, you just apply the addtion formulas for two angles being the same.

sin(2A) = sin(A + A) = sinAcosA + sinAcosA = 2sinAcosA
cos(2A) = cos(A + A) = cosAcosA - sinAsinA = cos^2(A) - sin^2(A)
remember, sin^2(A) + cos^2(A) = 1
so sin^2(A) = 1 - cos^2(A) and,
cos^2(A) = 1 - sin^2(A)

so cos(2A) = 1 - sin^2(A) - sin^2(A) = 1 - 2sin^2(A)
or cos(2A) = cos^2(A) - (1 - cos^2(A)) = 2cos^2(A) - 1

tan(2A) = 2tanA/(1 - tan^2(A))

Originally Posted by Pajkaj

My teacher was sick the day we learned the trig identities in class and I can't seem to find them in my text book - can someone post the oh say ten or so basic ones I should for sure know? Double angle functions, pythag ones, etc.?

for pythogorian identities, see diagram.

sin(x) = opp/hyp
cos(x) = adj/hyp
tan(x) = opp/adj = sin(x)/cos(x)

sec(x) = 1/cos(x) = hyp/adj
csc(x) = 1/sin(x) = hyp/opp
cot(x) = 1/tan(x) = cos(x)/sin(x) = adj/opp


a mnemonic to remember these (which I never use but other people seem to find it helpful) is:
SOHCAHTOA
meanng
Sine is Opposite over Hypotenuse Cosine is Adjacent over Hypotenuse Tangent is Oposite over Adjacent