Math 220
January 22
I. Are the following functions even, odd or neither?
1. f (x) = (ex + e−x )3
2. f (x) = (e−x + ex )2
3. f (x) = sin(x2 )
4. f (x) = cos(x3 )
5. f (x) = tan(x)
6. f (x) = csc(x) + sin(x)
7. f (x) = sec(x) + cot(x)
II. Solve for a and b.
1. 6 sin2 (x) + 3 cos2 (x) = a cos2 (x) + b
2. 5 tan2 (x) + 4 sec2 (x) = a tan2 (x) + b
3. 7 csc2 (x) + 3 cot2 (x) = a cot2 (x) + b
4. 3 sin2 (x) + 2 = a cos2 (x) + b sin2 (x)
III. Simiply the following expressions:
1. sin(π + θ) + sin(θ)
2. sin(π/2 + θ) + cos(θ)
3. sin(π/2 − θ) + cos(θ)
1
4. cos(π/2 + θ) + sin(θ)
5. cos(π/2 − θ) + sin(θ)
6. cot2 (θ + 5π/6) − csc2 (θ + 5π/6)
7. cos2 (π + θ) + sin2 (θ − π)
IV. Graph the following functions:
1. y = 2 tan(−x)
2. y = sec(−x)
3. y = − csc(x + π)
4. y = sec(x + π/2)
5. y = sec(π/2 − x)
6. y = 3 tan(x/2 + π) + 1
1
Solutions
I. Are the following functions even, odd or neither?
1. f (x) = (ex + e−x )3
Answer:
f (−x) = (e−x + e−(−x) )3
= (ex + e−x )3
= f (x)
f (x) is even.
2
2. f (x) = (e−x + e−(−x) )2
Answer:
f (−x) = (e−x + e−(−x) )2
= (ex + e−x )2
= f (x)
f (x) is even.
3. f (x) = sin(x2 )
Answer:
f (−x) = sin((−x)2 )
= sin(x2 )
f (x) is even.
4. f (x) = cos(x3 )
Answer:
f (−x) = cos((−x)3 )
= cos(−x3 )
= cos(x3 )
f (x) is even.
5. f (x) = tan(x)
3
Answer:
f (−x) = tan(−x)
sin(−x)
cos(−x)
− sin(x)
=
cos(x)
sin(x)
=−
cos(x)
= − tan(x)
=
f (x) is odd.
6. f (x) = csc(x) + sin(x)
Answer:
f (−x) = csc(−x) + sin(−x)
1
=
+ sin(−x)
sin(−x)
1
− sin(x)
=−
sin(x)
= − csc(x) − sin(x)
= −f (x)
f (x) is odd.
7. f (x) = sec(x) + cot(x)
Answer:
4
f (−x) = sec(−x) + cot(−x)
1
cos(−x)
+
cos(−x)
sin(−x)
1
cos(x)
+
=
cos(x) − sin(x)
= sec(x) − cot(x)
=
6= −f (x) or f (x)
f (x) is neither.
II. Solve for a and b.
1. 6 sin2 (x) + 3 cos2 (x) = a cos2 (x) + b
Answer:
Use sin2 (x) + cos2 (x) = 1
6 sin2 (x) + 3 cos2 (x) = 6(1 − cos2 (x)) + 3 cos2 (x)
= 6 − 6 cos2 (x) + 3 cos2 (x)
= 6 − 3 cos2 (x)
a = −3, b = 6
2. 5 tan2 (x) + 4 sec2 (x) = a tan2 (x) + b
Answer:
Use tan2 (x) + 1 = sec2 (x)
5 tan2 (x) + 4 sec2 (x) = 5 tan2 (x) + 4(tan2 (x) + 1)
= 5 tan2 (x) + 4 tan2 (x) + 4
= 9 tan2 (x) + 4
a = 9, b = 4
3. 7 csc2 (x) + 3 cot2 (x) = a cot2 (x) + b
Answer:
5
Use 1 + cot2 (x) = csc2 (x)
7 csc2 (x) + 3 cot2 (x) = 7(1 + cot2 (x)) + 3 cot2 (x)
= 7 + 7 cot2 (x) + 3 cot2 (x)
= 7 + 10 cot2 (x)
a = 10, b = 7
4. 3 sin2 (x) + 2 = a cos2 (x) + b sin2 (x)
Answer:
Use 1 = sin2 (x) + cos2 (x)
3 sin2 (x) + 2 = 3 sin2 (x) + 2(sin2 (x) + cos2 (x))
= 5 sin2 (x) + 2 cos2 (x)
a = 2, b = 5
III. Simiply the following expressions:
1. sin(π + θ) + sin(θ)
Answer:
Use sin(π + θ) = − sin(θ)
sin(π + θ) + sin(θ) = − sin(θ) + sin(θ)
=0
2. sin(π/2 + θ) + cos(θ)
Answer:
Use sin(π/2 + θ) = cos(θ)
sin(π/2 + θ) + cos(θ) = cos(θ) + cos(θ)
= 2 cos(θ)
3. sin(π/2 − θ) + cos(θ)
Answer:
6
Use sin(π/2 − θ) = − cos(θ)
sin(π/2 − θ) + cos(θ) = − cos(θ) + cos(θ)
=0
4. cos(π/2 + θ) + sin(θ)
Answer:
Use cos(π/2 + θ) = − sin(θ)
cos(π/2 + θ) + sin(θ) = − sin(θ) + sin(θ)
=0
5. cos(π/2 − θ) + sin(θ)
Answer:
Use cos(π/2 − θ) = − sin(θ)
cos(π/2 − θ) + sin(θ) = − sin(θ) + sin(θ)
=0
6. cot2 (θ + 5π/6) − csc2 (θ + 5π/6)
Answer:
Use 1 + cot2 (x) = csc2 (x) =⇒ cot2 (x) − csc2 (x) = −1
cot2 (θ + 5π/6) − csc2 (θ + 5π/6) =
= −1
7. cos2 (π + θ) + sin2 (θ − π)
Answer:
Use cos(π + θ) = − cos(θ) and sin(θ − π) = − sin(θ)
cos2 (π + θ) + sin2 (θ − π) = (− cos(θ))2 + (− sin(θ))2
= cos2 (θ) + sin2 (θ)
=1
IV. Graph the following functions:
7
1. y = 2 tan(−x)
Answer:
10
5
-6
-4
-2
2
4
6
2
4
6
2
4
6
-5
-10
2. y = sec(−x)
Answer:
6
4
2
-6
-4
-2
-2
-4
-6
3. y = − csc(x + π)
Answer:
5
-6
-4
-2
-5
4. y = sec(x + π/2)
8
Answer:
6
4
2
-6
-4
-2
2
4
6
2
4
6
2
4
6
-2
-4
-6
5. y = sec(π/2 − x)
Answer:
5
-6
-4
-2
-5
6. y = 3 tan(x/2 + π) + 1
Answer:
20
10
-6
-4
-2
-10
9