Download 7.1 - Math TAMU

1
c
Marcia
Drost, March 10, 2008
150 Lecture Notes - Section 7.1
Trigonometric Identities
5
Given csc θ = ,
3
tan θ =
3
4
Find the other four trigonometric functions.
1
Solution: csc u =
sin u
1
1
3
sin u =
= 5 =
csc u
5
3
since
sin2 u + cos2 u = 1
2
3
+ cos2 u = 1
5
cos2 u = 1 −
cos2 u =
9
25
16
25
cos u = ±
4
5
Since sin u > 0 and tan u > 0, u is in QI.
Therefore cos u =
4
5
sin u =
3
5
csc u =
5
3
cos u =
4
5
sec u =
5
4
tan u =
3
4
cot u =
4
3
Simplify: sin x cos2 x − sin x
sin x(cos2 x − 1)
sin x(− sin2 x)
− sin3 x
Since sin2 x + cos2 x = 1
cos2 x − 1 = − sin2 x
2
c
Marcia
Drost, March 10, 2008
Simplify:
sin θ
cos θ
+
cos θ 1 + sin θ
Verify the identity:
cos θ
sin θ
+
= csc θ
1 + cos θ sin θ
(sin θ)(sin θ) + (cos θ)(1 + cos θ)
=
(1 + cos θ)(sin θ)
sin2 θ + cos θ + cos2 θ
=
(1 + cos θ)(sin θ)
sin2 θ + cos2 θ + cos θ
=
(1 + cos θ)(sin θ)
1 + cos θ
=
(1 + cos θ)(sin θ)
1
=
sin θ
csc θ = csc θ
Factoring expressions:
tan2 x − tan2 x · sin2 x
tan2 x (1 − sin2 x)
tan2 x (cos2 x)
sin2 x
· cos2 x
2
cos x
sin2 x
Factor:
4 tan2 θ + tan θ − 3
(4 tan θ − 3) (tan θ + 1)
Factor:
csc3 x − csc2 x − csc x + 1
csc2 x(csc x − 1) − 1(csc x − 1)
(csc x − 1) (csc2 x − 1)
(csc x − 1) (csc x − 1) (csc x + 1)
c
Marcia
Drost, March 10, 2008
3
Simplify:
sin t + cot t · cos t
cos t
· cos t
sin t
cos2 t
sin t +
sin t
2
sin t cos2 t
+
sin t
sin t
2
sin t + cos2 t
1
=
= csc t
sin t
sin t
sin t +
Rewrite:
1
so that it is not in fractional form.
1 + sin x
1
1
1 − sin x
=
·
1 + sin x
1 + sin x 1 − sin x
1 − sin x
=
1 − sin2 x
1 − sin x
=
cos2 x
1
sin x
=
−
cos2 x cos2 x
sin x
1
1
−
·
=
2
cos x cos x cos x
= sec2 x − tan x · sec x
Rewrite:
1
1
−
so that it is not in fractional form.
1 − sin x 1 + sin x
1
1
−
, start with the
1 − sin x 1 + sin x
RHS (right hand side) and simplify till it matches the LHS (left hand side).
Note: To verify the identity 2 tan x · sec x =
Verify:
cos u
= sec u + tan u
1 − sin u
4
c
Marcia
Drost, March 10, 2008
Use x = 2 tan θ, 0 < θ <
√
√
π
, to express 4 + x2 as a trigonometric function of θ.
2
p
4 + (2 tan θ)2
√
= 4 + 4 tan2 θ
p
= 4(1 + tan2 θ)
p
= 4 sec2 θ)
4 + x2 =
= 2 sec θ
Fundamental Trigonometric Identities
Reciprocal Identities
csc x =
1
sin x
sec x =
tan x =
1
cos x
sin x
cos x
cot x =
cot x =
1
tan x
cos x
sin x
Pythagorean Identities
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Even-Odd Identities
sin(−x) = − sin x
cos(−x) = cos x
tan(−x) = − tan x
tan(90o − u) = cot u
sec(90o − u) = csc u
Cofunction Identities
sin(90o − u) = cos u
cos(90o − u) = sin u
cot(90o − u) = tan u
csc(90o − u) = sec u