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cos. is the x-coordinate of the point. sin. is the y-coordinate of the point. The picture of the unit circle and these coordinates looks like this: Some trigonometric identities follow immediately from this de nition, in particular, since the unit circle is all the points in plane with x and y coordinates satisfying x2 + y2 = 1, we have cos2.
cos(α−β) = cosαcosβ +sinαsinβ. From this we can derive expressions for cos( α + β ), sin( α + β ) and sin( α−β ). In order to do this we need to know the following results:
Fundamental trig identity. cos(. (cos x)2 + (sin x)2 = 1. 1 + (tan x)2 = (sec x)2 (cot x)2 + 1 = (cosec x)2.
Basic Identities. The functions cos(θ) and sin(θ) are defined to be the x and y coordinates of the point at an angle of θ on the unit circle. Therefore, sin(−θ) = − sin(θ), cos(−θ) = cos(θ), and sin2(θ) + cos2(θ) = 1. The other trigonometric functions are defined in terms of sine and cosine: tan(θ) = sin(θ)/ cos(θ) cot(θ ...
What is an identity? How do we verify an identity? Consider the trigonometric equation \ (\sin (2x) = \cos (x)\). Based on our current knowledge, an equation like this can be difficult to solve exactly because the periods of the functions involved are different.
Trigonometric Identities. sin2x+cosx=1 1+tan2x= secx. 1+cot2x= cscx. sinx=cos(90−x) =sin(180−x) cosx=sin(90−x) = −cos(180−x) tanx=cot(90−x) = −tan(180−x) Angle-sum and angle-difference formulas. sin(a± b) =sinacosb± cosasinb cos(a± b) =cosacosbmsinasinb tan( ) tan tan tan tan. a b a b a b. ± = ± 1m cot( ) cot cot cot cot.
The “big three” trigonometric identities are sin2 t+cos2 t = 1 (1) sin(A+B) = sinAcosB +cosAsinB (2) cos(A+B) = cosAcosB −sinAsinB (3) Using these we can derive many other identities. Even if we commit the other useful identities to memory, these three will help be sure that our signs are correct, etc. 2 Two more easy identities
