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3.3 Β· P on Circle
A point P is at angle ΞΈ on a circle of radius r. What are the coordinates of P?
cos ΞΈ = x/r and sin ΞΈ = y/r
Multiply both sides by r
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3.3 Β· Axis Values
Give all 8 axis values: cos and sin for 0, Ο/2, Ο, and 3Ο/2
The 4 axis points (1,0) (0,1) (β1,0) (0,β1)
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3.3 Β· Deriving Ο/6
Derive cos(Ο/6) and sin(Ο/6) using an equilateral triangle.
y = 1/2, use xΒ² + yΒ² = 1
Q1 so x is positive
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3.3 Β· Deriving Ο/4
Derive cos(Ο/4) and sin(Ο/4) using an isosceles right triangle.
x = y, use xΒ² + xΒ² = 1
Both values are equal
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3.3 Β· Ο/3 Family
Give cos and sin for Ο/3. Then state coordinates at 2Ο/3, 4Ο/3, and 5Ο/3.
Q1 values then apply ASTC signs
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3.3 Β· Symmetry β ASTC
How do the signs of sin and cos change across the four quadrants?
All Students Take Calculus
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3.3 Β· Ex 7 β Q2 angle
Find sin(2Ο/3) and cos(2Ο/3).
2Ο/3 is in Q2. Ref angle = Ο/3.
Q2: sin positive, cos negative
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3.3 Β· Ex 7 β Q3 angle
Find cos(4Ο/3) and sin(7Ο/6).
Both in Q3. Ref angles: Ο/3 and Ο/6.
Q3: both negative
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✓ Axis Values (all 8)
cos0=1 sin0=0 Β· cos(Ο/2)=0 sin(Ο/2)=1 Β· cosΟ=β1 sinΟ=0 Β· cos(3Ο/2)=0 sin(3Ο/2)=β1
Points: (1,0) (0,1) (β1,0) (0,β1). cos=x-coord, sin=y-coord. (Example 3.)
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✓ P on Circle Formula
P = (r cosΞΈ, r sinΞΈ)
From cos=x/r and sin=y/r: multiply by r. Unit circle (r=1): P = (cosΞΈ, sinΞΈ).
4/10
✓ Deriving Ο/4
cos(Ο/4) = β2/2 and sin(Ο/4) = β2/2
Isosceles right: x=y. Then 2xΒ²=1 β x=1/β2=β2/2. Both sin and cos equal β2/2.
3/10
✓ Deriving Ο/6
cos(Ο/6) = β3/2 and sin(Ο/6) = 1/2
Equilateral triangle: y=1/2. Then xΒ²+(1/2)Β²=1 β x=β3/2 (positive in Q1).
6/10
✓ Symmetry β ASTC Signs
Q1:(+,+) Q2:(β,+) Q3:(β,β) Q4:(+,β) [cos, sin]
All Students Take Calculus. All positive in Q1. Sin+ in Q2. Tan+ in Q3. Cos+ in Q4.
5/10
✓ Ο/3 Family
Ο/3:(1/2,β3/2) 2Ο/3:(β1/2,β3/2) 4Ο/3:(β1/2,ββ3/2) 5Ο/3:(1/2,ββ3/2)
Reference angle Ο/3 in all four quadrants. Apply ASTC signs.
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✓ Ex 7 β Q3 Angles
cos(4Ο/3) = β1/2 and sin(7Ο/6) = β1/2
Q3: both negative. 4Ο/3 ref Ο/3: cos=β1/2. 7Ο/6 ref Ο/6: sin=β1/2. (Ex 7e,f.)
7/10
✓ Ex 7 β Q2 Angle
sin(2Ο/3) = β3/2 and cos(2Ο/3) = β1/2
Q2: sin positive, cos negative. Ref angle Ο/3 β |sin|=β3/2, |cos|=1/2. (Ex 7a, 7i.)
9/10
3.3 Β· Ex 7 β Q4 angle
Find cos(11Ο/6) and sin(7Ο/4).
11Ο/6 in Q4 (ref Ο/6). 7Ο/4 in Q4 (ref Ο/4).
Q4: cos positive, sin negative
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3.3 Β· Ex 7 β Special
Find cos(5Ο/3) and sin(3Ο/2).
5Ο/3 is in Q4. 3Ο/2 is on the βy axis.
10/10
✓ Ex 7 β Special Values
cos(5Ο/3) = 1/2 and sin(3Ο/2) = β1
5Ο/3: Q4, ref Ο/3, cos=+1/2. 3Ο/2: bottom of unit circle (0,β1), sin=β1. (Ex 7j, 7h.)
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✓ Ex 7 β Q4 Angles
cos(11Ο/6) = β3/2 and sin(7Ο/4) = ββ2/2
Q4: cos positive, sin negative. 11Ο/6 ref Ο/6: cos=β3/2. 7Ο/4 ref Ο/4: sin=ββ2/2. (Ex 7c, 7d.)