Download Section 6.1: Verifying Trigonometric Identities

Section 6.1: Verifying Trigonometric Identities
Lecture 28
– Typeset by FoilTEX –
Outline
• Definition
Identities
of
Trigonometric
• Fundamental Identities
• Examples
(x,y)
t
Trigonometric Identities
• Let f (t), g(t) be functions which involve some (or all) of the six basic
trigonometric functions: sin(t), cos(t), tan(t), csc(t), sec(t), cot(t).
• A trigonometric identity is an equation of the form f (t) = g(t) which
holds true for all values of t in the domain of f and g.
∗ tan(t) =
sin(t)
cos(t)
is a trigonometric identity.
∗ sin(t) = cos(t) is not a trigonometric identity.
Fundamental Identities
• Reciprocal Identities:
1
csc(t) =
sin(t)
1
sec(t) =
cos(t)
1
cot(t) =
tan(t)
• Tangent and Cotangent Identities:
tan(t) =
sin(t)
cos(t)
cot(t) =
cos(t)
sin(t)
Fundamental Identities
• Pythagorean Identities:
∗ sin2(t) + cos2(t) = 1
∗ 1 + tan2(t) = sec2(t)
∗ 1 + cot2(t) = csc2(t)
Verifying Identities
• There are a couple of ways to verify (or prove) a trigonometric identity
f (t) = g(t).
1) Use the fundamental identities and algebra to transform the left
hand side into the right hand side (or vice versa).
2) First use fundamental identities and algebra to transform the
left hand side into some equivalent expression h(t). Then use
fundamental identities and algebra to transform the right hand side
into h(t).
Example
• Verify the identity sec(t) = sin(t)(tan(t) + cot(t)).
Example
• Verify the identity
1 + sin(x)
cos(x)
=
.
1 − sin(x)
cos(x)
Example
• Verify the identity
(tan θ − sec θ)2 =
1 − sin θ
.
1 + sin θ