4.5 (day 1) inverse sine & cosine. remember: the inverse of a function is found by switching the...
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4.5 (Day 1) Inverse Sine & Cosine
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Remember: the inverse of a function is found by switching the x & y values (reflect over line y = x)
Domains become ranges…. ranges become domains
We want the inverse sine & inverse cosine to be functions(pass vertical line test) so we need to restrict their domains –can’t be all real numbers
To denote we want the inverses to be functions, we use capital letters Sin–1x and Cos–1x
OR Arcsin x and Arccos x
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y = Sin–1 x
D = [–1, 1]
Want range to include (–) & (+) valuesChoose QI for (+) valuesWhich quad is closest to QI that contains (–) values?
y = Cos–1 x
D = [–1, 1](the range for sin θ)
(the range for cos θ)
III
III IV
(+)(+)
(–) (–)
,2 2
R
close
III
III IV
(–)
(–) (+)
0,R
close
(+)
These re
strict
ions tell u
s where
we draw
the r
eferen
ce θ &
△
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*What type of answer is required
(1) sin x cos x
(2) Sin–1 xCos–1 x
trig function of an angle is a ratiotrig (angle) = ratio
a) Sin–1 0.3240
0.3300
b) Arcsin 0.5681
0.6042
Ex 1) Evaluate to 4 decimal places (radian mode)
inverse function of a ratio is an angletrig–1 (ratio) = angle Find the θ and draw it’s picture in the correct quadrant!
c) Cos–1 (–0.56)
2.1652
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short △ QIII
Ex 2) Evaluate. Find exact value if possible.
a) 1
1
Sin sin4
2Sin
2 4
b)
22ratio
θ2
21
1
Cos cos2
Cos (0)2
ratio 0
c) 7Arcsin sin
6
1Arcsin
2 6
12ratio
θ
tall △ QIV
c) 5
Arccos cos3
1Arcsin
2 3
12ratio
θ
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picture
Ex 3) Determine the exact value.
a) 1 5sin Cos
9
2 14sin
9
θ θ
56 2 14
picture
b) 3cos Arcsin
5
4cos
5
θ θ
9
5
(a ratio!)
–35
4
(a ratio!)
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Optics: Light is refracted when it travels from air to water. i is the angle of incidence (in air) and r is the angle of refraction (in water). Equation is:
Ex 4) If a light ray makes a 30° angle with the vertical in air, determine the angle with the vertical in water.
*Degree mode*
sin 4
sin 3
i
r
12
1
sin30 4 4
sin 3 sin 33 3
4sin sin2 83
Sin8
22
r r
r r
r
r
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Homework
#405 Pg 220 #1–13 all, 15–18, 22, 25, 28–31