Download Section 1.4: Trig Functions and Their Inverses

Section 1.4: Trig Functions and Their Inverses
Homework Questions
π
:
6
π
1
Solution: sin =
6
2
1. Evaluate sin
5π
2. Evaluate: tan −
6
Solution:
5π
5π
sin
sin −
5π
6
6
=−
tan −
=
5π
5π
6
cos −
cos
6
6
1/2
=− √
− 3/2
√
3
=
3
3. Establish the identity (1 + cot2 θ) sin2 θ = 1
Solution:
(1 + cot2 θ) sin2 θ = sin2 θ cot2 θ sin2 θ
cos2 θ
sin2 θ
sin2 θ
= sin2 θ + cos2 θ = 1
= sin2 θ +
4. Find the exact value of sin−1
(1)
√ !
3
2
Solution:
The question is asking at what angle θ is sin θ =
1
√
3/2?. We want to solve
−1
sin
√ !
3
= θ for θ. This gives
2
√ !!
3
= sin θ
2
−1
sin sin
or
√
sin θ =
π
3
so θ =
2
3
5. Find the exact value of sec−1 (−1).
Solution: We know that sec θ =
sec
−1
1
, so
cos θ
−1
(−1) = cos
1
−1
=π
6. Use a right triangle to write cos(sin−1 (5x)) as an algebraic expression. Assume x is positive and that the given inverse trigonometric is defined for x.
Solution: We know that sin θ = 5x. The corresponding right triangle has
opposite side length 5x and hypotenuse
length 1. By the Pythagorean Theorem,
√
the adjacent side has length 1 − 25x2 . We know that cos θ is Adjacent over
Hypotenuse, giving
√
1 − 25x2
−1
cos(sin (5x)) =
1
7. Find the exact value for the remaining trigonometric functions of θ given
cos θ = −
24
, θ in quadrant III
25
Solution θ corresponds to the following right triangle:
2
but is in the third quadrant. So,
sin θ = −
7
25
7
24
24
cot θ =
7
25
sec θ = −
24
25
csc θ = −
7
tan θ =
8. Determine the amplitude and period of y = 5.4 cos
πx 9
2π
= 18
Solution: Amplitude = 5.4. Period =
πx/9
√
9. Solve the equation 2 cos x − 2 = 0 for all solutions of x.
√
√
2
Solution: 2 cos x − 2 = 0 gives cos x =
or
2
π
x = + 2πk,
4
x=
7π
+ 2πk
4
for all k = 0, 1, 2, . . .
10. Two ladders of length a lean against opposite walls of an alley with their
feet touching (see figure). One ladder extends h feet up the wall and makes a
75◦ angle with the ground. The other ladder extends k feet up the opposite wall
and makes a 45◦ angle with the ground. Find the width of the alley.
3
Solution: Width = a cos(75◦ ) + a cos(45◦ )
4