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Inverse Trig FunctionsLearning Goals:
1. Understand Domain/Range requirements for inverse trig functions
2. Be able to calculate exact values for inverse trig functions.
3. Understand the composition of inverse trig functions
4. Be able to calculate compositions with inverse trig functions.
Find Exact Values
• Get out that plate!• What is the angle whose sine is ½ ?• Arcsin( ½ ) = θ or sin-1( ½ ) = θ • What is the angle whose cosine is ½ ?• Arccos( ½ ) = θ or cos-1( ½ ) = θ • Are there other angles whose sine is
½ ?• Are there other angles whose cosine is
½?
x
y (0, 1)
90°2
(–1, 0)
180°
(0, –1)
270°32
360°2 (1, 0)0° 0
Unit CircleAnglesandCoordinates 60° 3
31 ,2 2
45°4
2 2,2 2
30° 6
3 1,2 2
330°116
3 1,2 2
315°74
2 2,2 2
300°53
31 ,2 2
23
120°
31 ,2 2
34
135°
2 2,2 2
56
150°
3 1,2 2
210°76
3 1,2 2
225°5
4
2 2,2 2
240°43
31 ,2 2
Everything shows up on
the next click.
If you can find values, you can make a graph!
• Sketchpad
Summary from Sketchpad
• With no restrictions, inverses of sin, cos and tangent are not functions.
• With restrictions:• Arcsin(x) = θ • Domain: {x|-1 ≤ x ≤ 1}• Range:{θ| -π/2
≤ θ ≤ π/2 }
• Arccos(x) = θ• Domain: {x|-1 ≤ x ≤ 1}• Range: {θ|0 ≤ θ ≤ π }• Arctan(x) = θ• Domain: {x|x є R}• Range: :{θ| -π/2
≤ θ ≤ π/2 }
So what?
• Q: What are you supposed to get out of knowing and understanding these domain restrictions?
• A: I hate answering questions on powerpoint slides. You give me the answer.
Compositions
• Y = sin x is the inverse of y = arcsin(x).
• So what is sin(arcsin(x))?
• Arcsin(sin(x))?
• See if you can find an example when these compositions wouldn’t work.
Practice Time
• Use your unit circle to do problems from the book: p. 465
x
y (0, 1)
90°2
(–1, 0)
180°
(0, –1)
270°32
360°2 (1, 0)0° 0
Unit CircleAnglesandCoordinates 60° 3
31 ,2 2
45°4
2 2,2 2
30° 6
3 1,2 2
330°116
3 1,2 2
315°74
2 2,2 2
300°53
31 ,2 2
23
120°
31 ,2 2
34
135°
2 2,2 2
56
150°
3 1,2 2
210°76
3 1,2 2
225°5
4
2 2,2 2
240°43
31 ,2 2
Everything shows up on
the next click.
