Download sinθ = y cosθ = x tanθ = y cotθ = x cscθ = r secθ = r csc sin θ θ = 1

Math 130
Verifying Trigonometric Identities
Definitions:
sin θ =
y
r
cosθ =
x
r
tan θ =
y
x
cot θ =
x
y
csc θ =
r
y
sec θ =
r
x
r
y
Reciprocal identities:
csc θ =
1
1
1
, sec θ =
, cot θ =
sin θ
cosθ
tan θ
θ
x
Negative angle identities:
sin( −θ ) = − sin θ ,cos( −θ ) = cosθ , tan( −θ ) = − tan θ
r
b
θ a
−θ
−b
r
Identities:
sin x
cos x
→ cot x =
cos x
sin x
2
2
2
2
2
2
2) sin x + cos x = 1 → tan x + 1 = sec x , 1 + cot x = csc x
1)
tan x =
TO DO
• Start with more complex side (more to work with)
• Always look for forms of known identities that you can apply (e.g.
1 − sin 2 x = cos2 x )
• Multiply denominators according to table below:
Denominator
Multiply by
1 ± sin x
1 m sin x
1 m sin x
1 ± cos x
1 m cos x
1 m cos x
sec x ± tan x
sec x m tan x
sec x m tan x
sec x ± 1
sec x m 1
sec x m 1
csc x ± cot x
csc x m cot x
csc x m cot x
csc x ± 1
csc x m 1
csc x m 1
• Multiply denominators of by
• Convert
tan x,cot x,csc x,sec x → sin x,cos x expressions then algebra, typically clearing fractions.
Multiply by conjugate only if it gets you a form equivalent to one of the Pythag. identities.
Otherwise convert to sin x,cos x .