Download Math 180 Chapter 6 Lecture Notes

Math 180
Chapter 6
Lecture Notes
Professor Miguel Ornelas
1
M. Ornelas
Math 180 Lecture Notes
Section 6.1
Verifying Trigonometric Identities
Verify the identity.
a.
sin x + cos x cot x = csc x
b.
(tan u + cot u)(cos u + sin u) = csc u + sec u
c.
1 + csc 3β
− cot 3β = cos 3β
sec 3β
d.
tan2 x
1 − cos x
=
sec x + 1
cos x
Section 6.1 continued on next page. . .
2
Section 6.1
M. Ornelas
e.
Math 180 Lecture Notes
cot θ − tan θ
= csc θ − sec θ
sin θ + cos θ
f.
1
= sin β cos β
tan β + cot β
b.
2 cos θ + 1 = 0
d.
(2 sin θ + 1)(2 cos θ + 3) = 0
Section 6.2
Trigonometric Equations
Find all solutions of the equation.
√
a.
sin x = −
2
2
√
2
1
c. cos x = −
4
2
Section 6.2 continued on next page. . .
3
Section 6.1 (continued)
M. Ornelas
Math 180 Lecture Notes
Find all solutions of the equation that are in the interval [0, 2π).
π
=0
a. cos 2x −
b. 2 cos2 t + 3 cos t = −1
4
c.
2 cos2 γ + cos γ = 0
d.
4
2 tan t − sec2 t = 0
Section 6.2 (continued)
M. Ornelas
Math 180 Lecture Notes
Section 6.3
The Addition and Subtraction Formulas
Cofunction Formulas
π
1. cos
− u = sin u
2
3.
tan
5.
sec
π
2
π
2
2.
sin
− u = cot u
4.
cot
− u = csc u
6.
csc
π
2
π
2
π
2
− u = cos u
− u = tan u
− u = sec u
Addition and Subtraction Formulas for Cosine, Sine and Tangent
1. cos(u − v) = cos u cos v + sin u sin v
2. cos(u + v) = cos u cos v − sin u sin v
3. sin(u + v) = sin u cos v + cos u sin v
4. sin(u − v) = sin u cos v − cos u sin v
5. tan(u + v) =
tan u + tan v
1 − tan u tan v
6. tan(u − v) =
tan u − tan v
1 + tan u tan v
Express as a cofunction of a complementary angle.
a.
cos
π
3
Section 6.3 continued on next page. . .
b.
5
tan 1
Section 6.3
M. Ornelas
Math 180 Lecture Notes
Find the exact values.
a.
cos
π
π
+ cos
4
6
b.
tan 60◦ + tan 225◦
b.
sin(−5) cos 2 + cos 5 sin(−2)
Express as a trigonometric function of one angle.
a.
cos 61◦ sin 82◦ − sin 61◦ cos 82◦
If sin α = −
π
5
and tan α > 0, find the exact value of sin α −
13
3
Section 6.3 continued on next page. . .
6
Section 6.3 (continued)
M. Ornelas
Math 180 Lecture Notes
If α and β are acute angles such that csc α =
a.
sin(α + β)
b.
Section 6.3 (continued)
13
4
and cot β = , find
12
3
tan(α + β)
c.
the quadrant containing α + β
Verify the identity.
a.
π 1 + tan u
=
tan u +
4
1 − tan u
b.
7
sin(u + v) · sin(u − v) = sin2 u − sin2 v
M. Ornelas
Math 180 Lecture Notes
Section 6.4
Multiple-Angle Formulas
Double-Angle Formulas
1. sin 2u = 2 sin u cos u
2. cos 2u = cos2 u − sin2 u
3. cos 2u = 1 − 2 sin2 u
4. tan 2u =
If sin α =
2 tan u
1 − tan2 u
4
and α is an acute angle, find the exact values of sin 2α and cos 2α
5
Express cos 3θ in terms of cos θ
Section 6.4 continued on next page. . .
8
Section 6.4
M. Ornelas
Math 180 Lecture Notes
Half-Angle Identities
1. sin2 u =
1 − cos 2u
2
2. cos2 u =
1 + cos 2u
2
3. tan2 u =
1 − cos 2u
1 + cos 2u
Verify the identity sin2 x cos2 x =
1
(1 − cos 4x).
8
Express cos4 t in terms of values of the cosine function with exponent 1.
Section 6.4 continued on next page. . .
9
Section 6.4 (continued)
M. Ornelas
Math 180 Lecture Notes
Half-Angle Formulas
r
v
1 − cos v
1. sin = ±
2
2
v
2. cos = ±
2
r
1 + cos v
2
v
3. tan = ±
2
r
1 − cos v
1 + cos v
Find exact values for
a.
sin 22.5◦
b.
Half-Angle Formulas for the Tangent
1. tan
v 1 − cos v
=
2
sin v
2. tan
v
sin v
=
2 1 + cos v
α
4
If tan α = − and α is in quadrant IV, find tan .
3
2
10
cos 112.5◦
Section 6.4 (continued)
M. Ornelas
Math 180 Lecture Notes
Section 6.5
Product-to-Sum and Sum-to-Product Formulas
Product-to-Sum Formulas
1. sin u cos v =
1
[sin(u + v) + sin(u − v)]
2
2. cos u sin v =
1
[sin(u + v) − sin(u − v)]
2
3. cos u cos v =
1
[cos(u + v) + cos(u − v)]
2
4. sin u sin v =
1
[cos(u − v) − cos(u + v)]
2
Express as a sum or difference.
a.
b.
sin 7t sin t
Sum-to-Product Formulas
1. sin a + sin b = 2 sin
a+b
a−b
cos
2
2
2. sin a − sin b = 2 cos
a+b
a−b
sin
2
2
3. cos a + cos b = 2 cos
a+b
a−b
cos
2
2
4. cos a − cos b = −2 sin
a+b
a−b
sin
2
2
Section 6.5 continued on next page. . .
11
3 cos x sin 2x
Section 6.5
M. Ornelas
Math 180 Lecture Notes
Express as a product.
a.
sin 2θ − sin 8θ
b.
cos x + cos 2x
b.
1
sin u − sin v tan 2 (u − v)
=
sin u + sin v tan 12 (u + v)
Verify the identity.
a.
sin 4t + sin 6t
= cot t
cos 4t − cos 6t
Use sum-to-product formulas to find the solutions of the equation.
a.
sin 5t + sin 3t = 0
b.
12
cos 3x = − cos 6x
Section 6.5 (continued)
M. Ornelas
Math 180 Lecture Notes
Section 6.6
The Inverse Trigonometric Functions
Definition of the Inverse Sine Function
The inverse sine function, denoted by sin−1 , is defined by
y = sin−1 x if and only if x = sin y
for −1 ≤ x ≤ 1 and −
π
π
≤y≤ .
2
2
Definition of the Inverse Cosine Function
The inverse cosine function, denoted by cos−1 , is defined by
y = cos−1 x if and only if x = cos y
for −1 ≤ x ≤ 1 and 0 ≤ y ≤ π.
Definition of the Inverse Tangent Function
The inverse tangent function, denoted by tan−1 , is defined by
y = tan−1 x if and only if x = tan y
for any real number x and for −
π
π
<y< .
2
2
Find the exact value of the expression whenever it is defined.
−1
√ 
 2 


2
a.
sin
c.
√ tan−1 − 3
Section 6.6 continued on next page. . .
π
2
b.
cos−1
d.
arccos
13
π
3
Section 6.6
M. Ornelas
Math 180 Lecture Notes
!#
"
3
e. sin arcsin −
10
7π
g. arctan tan
4
−1
Section 6.6 (continued)
!#
"
5π
cos
6
f.
cos
h.
tan cos−1 0
!
"
!
#
3
4
i. cos arctan − − arcsin
4
5
j.
Section 6.6 continued on next page. . .
14
−1
tan 2 tan
3
4
!
M. Ornelas
Math 180 Lecture Notes
Section 6.6 (continued)
Write the expression as an algebraic expression in x for x > 0.
a.
sin tan−1 x
b.
x
csc tan−1
2
Use inverse trigonometric functions to find the solutions of the equation cos2 x + 2 cos x − 1 = 0 that are in the interval
[0, 2π).
15