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3.9 · Notation
Write sin(π/3) = √3/2 in both inverse trig notations.
What does the inverse swap?
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3.9 · Output of arcsin
What type of value does arcsin(x) always output?
Not a ratio — something else
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3.9 · arcsin Range
What is the range of sin⁻¹(x)? Which quadrants does it cover?
Think: where is sin restricted to make it 1-to-1?
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3.9 · arccos Range
What is the range of cos⁻¹(x)? Which quadrants does it cover?
Different from arcsin!
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3.9 · arctan Range
What is the range of tan⁻¹(x)? Why are the brackets open (not closed)?
Think about asymptotes
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3.9 · Evaluate a
Evaluate: cos⁻¹(−√2/2)
Range of arccos = [0,π]
Negative input → which quadrant for arccos?
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3.9 · Evaluate b
Evaluate: sin⁻¹(−√3/2)
Range of arcsin = [−π/2, π/2]
Negative input → which quadrant for arcsin?
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3.9 · Evaluate c
Evaluate: tan⁻¹(√3)
Range of arctan = (−π/2, π/2)
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✓ Output is Always an Angle
arcsin(x) outputs an ANGLE (in radians). It is the angle whose sine equals x. Never a ratio — always an angle measure.
arcsin(1/2) = π/6: the angle whose sine is 1/2. arccos(−1) = π: the angle whose cosine is −1.
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✓ Notation
arcsin(√3/2) = π/3 OR sin⁻¹(√3/2) = π/3
sin(π/3) = √3/2 swaps to: 'the angle with sine √3/2 is π/3'. Both notations are identical — read aloud as 'arcsine of √3/2 equals π/3'.
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✓ arccos Range
Range = [0, π] · Covers Q1 (0 to π/2) and Q2 (π/2 to π)
arccos can NEVER output a negative angle. Negative input → Q2 answer (between π/2 and π). Positive input → Q1 answer (between 0 and π/2).
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✓ arcsin Range
Range = [−π/2, π/2] · Covers Q4 (−π/2 to 0) and Q1 (0 to π/2)
Negative input → negative angle (Q4). Positive input → positive angle (Q1). The range is symmetric around 0.
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✓ Evaluate cos⁻¹(−√2/2)
= 3π/4
Reference angle: cos = √2/2 at π/4. Input is negative → arccos → Q2. Q2 angle = π − π/4 = 3π/4. Check: cos(3π/4) = −√2/2 ✓
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✓ arctan Range
Range = (−π/2, π/2) · Open brackets — ±π/2 excluded · Covers Q4 and Q1
Open brackets because tan is undefined at ±π/2 (vertical asymptotes). Those values cannot be outputs of arctan.
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✓ Evaluate tan⁻¹(√3)
= π/3
tan(π/3) = √3 (from unit circle: (√3/2)÷(1/2)=√3). Input positive → Q1 → π/3. Check: π/3 is in (−π/2, π/2) ✓
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✓ Evaluate sin⁻¹(−√3/2)
= −π/3
Reference angle: sin = √3/2 at π/3. Input is negative → arcsin → Q4 → negative angle = −π/3. Check: sin(−π/3) = −√3/2 ✓
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3.9 · Reflection P
P=(x,y) in Q1. What are the coordinates of Q (y-axis reflection), R (origin), S (x-axis)?
Which coordinate flips in each case?
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3.9 · Reflection Match
P=(x,y) in Q1. Which point does cos⁻¹(−x) intersect? Which does sin⁻¹(−y) intersect?
cos range [0,π] · sin range [−π/2,π/2]
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✓ Reflection Match
cos⁻¹(−x) → Q (Q2, coords (−x,y)) · sin⁻¹(−y) → S (Q4, coords (x,−y))
cos⁻¹(−x): cos=−x, range [0,π] → Q2 → point (−x,y)=Q. sin⁻¹(−y): sin=−y, range [−π/2,π/2] → Q4 → point (x,−y)=S.
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✓ Reflection Points
Q=(−x,y) · R=(−x,−y) · S=(x,−y)
P=(x,y) in Q1. Q: y-axis reflection → x flips → (−x,y) in Q2. R: origin reflection → both flip → (−x,−y) in Q3. S: x-axis reflection → y flips → (x,−y) in Q4.