4.1 Derivatives of the Sine and Cosine Functions
Graphical reasoning can be applied to determine the exact derivative of a sinusoidal function.
Determine the derivative of sin x.
𝑥 𝑠𝑖𝑛 𝑥 𝑑
𝑑𝑦
𝑠𝑖𝑛 𝑥
0
fin co
π
I
V
2 O
π o 1
3π
2
1 0
2π
I
a) Complete the 𝑠𝑖𝑛 𝑥 column in the table.
b) Estimate the instantaneous rate of change (tangent slope) of 𝑠𝑖𝑛 𝑥 at 𝑥 = 0 using a secant line. Use the interval
0π 1π
⎡ 100 , ⎤.
⎣ 100 ⎦
I
0.999 1
c) Complete the instantaneous rate of change column.
𝑑
d) Graph the derivative function. What do you notice about the values of 𝑑𝑦
𝑠𝑖𝑛 𝑥? What is the derivative of 𝑠𝑖𝑛 𝑥?
Sina cossi
Determine the derivative of sin x.
𝑥 𝑐𝑜𝑠 𝑥 𝑑
𝑑𝑦
𝑐𝑜𝑠 𝑥
0
I
π
2
f sink
π
1 O
3π
2 O I
2π 0
a) Complete the 𝑐𝑜𝑠 𝑥 column in the table.
Graphical reasoning can be applied to determine the exact derivative of a sinusoidal function.
Determine the derivative of sin x.
𝑥 𝑠𝑖𝑛 𝑥 𝑑
𝑑𝑦
𝑠𝑖𝑛 𝑥
0
fin co
π
I
V
2 O
π o 1
3π
2
1 0
2π
I
a) Complete the 𝑠𝑖𝑛 𝑥 column in the table.
b) Estimate the instantaneous rate of change (tangent slope) of 𝑠𝑖𝑛 𝑥 at 𝑥 = 0 using a secant line. Use the interval
0π 1π
⎡ 100 , ⎤.
⎣ 100 ⎦
I
0.999 1
c) Complete the instantaneous rate of change column.
𝑑
d) Graph the derivative function. What do you notice about the values of 𝑑𝑦
𝑠𝑖𝑛 𝑥? What is the derivative of 𝑠𝑖𝑛 𝑥?
Sina cossi
Determine the derivative of sin x.
𝑥 𝑐𝑜𝑠 𝑥 𝑑
𝑑𝑦
𝑐𝑜𝑠 𝑥
0
I
π
2
f sink
π
1 O
3π
2 O I
2π 0
a) Complete the 𝑐𝑜𝑠 𝑥 column in the table.
