Math 1304 Calculus I
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Math 1304 Calculus I 3.2 – Derivatives of Trigonometric Functions
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Math 1304 Calculus I. 3.2 – Derivatives of Trigonometric Functions. Trigonometric Functions. Measure Radians, degrees Basic functions sin, cos, tan, csc, sec, cot Periodicity Special values at: 0, π /6, π /4, π /3, π /2, π Sign change Addition formulas Derivatives. Angle. - PowerPoint PPT Presentation
Transcript of Math 1304 Calculus I
Math 1304 Calculus I
3.2 – Derivatives of Trigonometric Functions
Trigonometric Functions
• Measure– Radians, degrees
• Basic functions– sin, cos, tan, csc, sec, cot
• Periodicity • Special values at:
– 0, π/6, π/4, π/3, π/2, π• Sign change• Addition formulas• Derivatives
Angle
• Radians: Measure angle by arc length around unit circle
θ
Definition of Basic Functions
• sin() = opposite / hypotenuse• cos() = adjacent / hypotenuse• tan() = opposite / adjacent • csc() = hypotenuse / opposite • sec() = hypotenuse / adjacent• cot() = adjacent / opposite
hypotenuse
opposite
adjacent
θ
Sin and Cos Give the Others
)sin()cos()cot(
)sin(1)csc(
)cos(1)sec(
)cos()sin()tan(
Sin, Cos, Tan on Unit Circle
θ
θ
cos(θ)
sin(θ)1
tan(θ)
)tan()sin(
Periodicity
)cot()2cot()csc()2csc()sec()2sec()tan()2tan()cos()2cos(
)sin()2sin(
Special Values
.1 /2)sin(
23 /3)sin(
22 /4)sin(
21 /6)sin(
0 sin(0)
.0 /2)cos(21 /3)cos(
22 /4)cos(
23 /6)cos(
1 cos(0)
Basic Inequalities
θ
θ
cos(θ)
sin(θ)1
tan(θ)
)tan()sin(
For 2/0
Proof of Basic Equalities
θ
θ
cos(θ)
sin(θ)1
tan(θ)
)tan(
EADE
EABE
D
E
A
B
)sin(
)sin( BABC
C
Draw tangent line at B.It intersects AD at E
O
Special Limit
1)sin(lim0
Use Squeezing Theorem
1/)sin()cos()cos(/1)sin(/1
)sin(/)tan()sin(/)sin(/)sin()tan()sin(
1)sin(lim1)sin(lim1
1lim)sin(lim)cos(lim
00
000
Another Special Limit
0)cos(1lim0
Addition Formulas
• sin(x+y) = sin(x) cos(y) + cos(x) sin(y)• cos(x+y) = cos(x) cos(y) – sin(x) sin(y)
Derivative of Sin and Cos
• Use addition formulas (in class)
Derivatives
• If f(x) = sin(x), then f’(x) = cos(x)• If f(x) = cos(x), then f’(x) = - sin(x)• If f(x) = tan(x), then f’(x) = sec2(x)• If f(x) = csc(x), then f’(x) = - csc(x) cot(x)• If f(x) = sec(x), then f’(x) = sec(x) tan(x)• If f(x) = cot(x), then f’(x) = - csc2(x)
A good working set of rules• Constants: If f(x) = c, then f’(x) = 0• Powers: If f(x) = xn, then f’(x) = nxn-1
• Exponentials: If f(x) = ax, then f’(x) = (ln a) ax
• Trigonometric Functions: If f(x) = sin(x), then f’(x)=cos(x) If f(x) = cos(x), then f’(x) = -sin(x) If f(x)= tan(x), then f’(x) = sec2(x) If f(x) = csc(x), then f’(x) = -csc(x) cot(x) If f(x)= sec(x), then f’(x) = sec(x)tan(x) If f(x) = cot(x), then f’(x) = -csc2(x)• Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x)• Sum: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x)• Difference: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x)• Multiple sums: derivative of sum is sum of derivatives• Linear combinations: derivative of linear combo is linear combo of derivatives• Product: If f(x) = g(x) h(x), then f’(x) = g’(x) h(x) + g(x)h’(x)• Multiple products: If F(x) = f(x) g(x) h(x), then F’(x) = f’(x) g(x) h(x) + …• Quotient: If f(x) = g(x)/h(x), then f’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2
