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3.5 Derivatives of Trig Functions
London Bridge, Lake Havasu City, Arizona
Photo by Vickie Kelly, 2001
Greg Kelly, Hanford High School, Richland, Washington
Consider the function
y  sin  
 slope
We could make a graph of the slope:



2
1
0
1
Now we connect the dots!
0

2
The resulting curve is a cosine curve.

1
0
d
sin  x   cos x
dx

We can do the same thing for y  cos  
 slope


The resulting curve is a sine curve that has
been reflected about the x-axis.

2
0

2

0
1
0
1
0
d
cos  x    sin x
dx

We can find the derivative of tangent x by using the
quotient rule.
d
tan x
dx
cos 2 x  sin 2 x
cos 2 x
d sin x
dx cos x
1
cos 2 x
cos x  cos x  sin x    sin x 
sec 2 x
cos 2 x
d
tan  x   sec2 x
dx

Derivatives of the remaining trig functions can be
determined the same way.
d
sin x  cos x
dx
d
cot x   csc 2 x
dx
d
cos x   sin x
dx
d
sec x  sec x  tan x
dx
d
tan x  sec 2 x
dx
d
csc x   csc x  cot x
dx

Clicker Quiz
Have paper ready, you will probably
need it.
A spring is bobbing up and down on the end of a
spring according to 𝑠 𝑡 = −3 sin 𝑡. Is the spring
3𝜋
traveling upwards or downwards at 𝑡 = ?
4
A. Upward
B. Downward
Which of the following is an equation of the
tangent line to 𝑦 = sin 𝑥 + cos 𝑥 at 𝑥 = 𝜋.
A.
B.
C.
D.
E.
𝑦 = −𝑥 + 𝜋 − 1
𝑦 = −𝑥 + 𝜋 + 1
𝑦 = −𝑥 − 𝜋 + 1
𝑦 = −𝑥 − 𝜋 − 1
𝑦 =𝑥−𝜋+1
Which of the following is an equation of the
normal line to 𝑦 = sin 𝑥 + cos 𝑥 at 𝑥 = 𝜋.
A.
B.
C.
D.
E.
𝑦 = −𝑥 + 𝜋 − 1
𝑦 =𝑥−𝜋−1
𝑦 =𝑥−𝜋+1
𝑦 =𝑥+𝜋+1
𝑦 =𝑥+𝜋−1
Find 𝑦′′ if 𝑦 = 𝑥 sin 𝑥
A. -x sin(x)
B. x cos(x) + sin(x)
C. -x sin(x) + 2cos(x)
D. x sin(x)
E. -sin(x) + cos(x)
A body is moving in simple harmonic motion
with position 𝑠 = 3 + sin 𝑡. At which of the
following times is the velocity of that body zero?
A. t=0
B. 𝑡 =
C. 𝑡 =
𝜋
4
𝜋
2
D. 𝑡 = 𝜋
E. none of these