Download Handy-Dandy Trig Identities for Calculus

Handy-Dandy Trig Identities for Calculus
June 2, 2015
Instructor: Maxx Cho
Math 141, Section 0101
Name:
1. The pythagorean identity:
sin2 x + cos2 x = 1
Divide both sides by sin2 x to get:
1 + cot2 x = csc2 x
Divide both sides by cos2 x to get:
tan2 x + 1 = sec2 x
2. The sum/difference formulas:
sin A ± B = sin A cos B ± cos A sin B
cos A ± B = cos A cos B ∓ sin A sin B
Note the difference between ± and ∓.
3. In the angle sum formulas in (2), you can set A = B and simplify to get the double-angle
formulas:
sin 2A = 2 sin A cos A
cos 2A = cos2 A − sin2 A
In the cosine double-angle identity, you can substitute with the pythagorean identity in (1) and
simplify to get:
cos 2A = 2 cos2 A − 1
cos 2A = 1 − 2 sin2 A
4. Take the last two cosine double angle formulas in (3), and solve them for cos2 A and sin2 A to
get the power-reduction formulas
cos 2A + 1
cos2 A =
2
1
−
cos
2A
sin2 A =
2
sin x
In (2), (3), and (4), you can use the fact that tan x = cos
x to derive the analogous formulas for
tan x as well.
5. Also, don’t forget some basic facts.
(1) sin x is an odd function. That means sin(−x) = − sin x.
(2) cos x is an even function. That means cos(−x) = + cos x.
(3) Since csc x = sin1 x , it is also odd.
(4) Since sec x = cos1 x , it is also even.
sin x
(5) Since tan x = cos
x , it is odd. So is cot x.
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