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AP Calculus BC
Unit 2: Differentiation: Definition and Fundamental Properties
2.10 Finding the Derivatives of Trigonometric Functions
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What are the two fundamental concepts that need to be reviewed before finding the derivatives of trigonometric functions?
Trigonometric identities and properties
The secant function is the reciprocal of the
cosine
function.
True
Trigonometric functions exhibit symmetry around the origin and the
y-axis
Steps to simplify trigonometric expressions using identities:
1️⃣ Identify the relevant identities
2️⃣ Apply the identities to simplify
3️⃣ Simplify further if needed
What are the bounded values of sine and cosine functions?
-1 and 1
The derivative of
sin
x
\sin x
sin
x
is \cos x
The quotient rule states that if f(x) = \frac{u(x)}{v(x)}</latex>, then
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
u
′
(
x
)
v
(
x
)
−
u
(
x
)
v
′
(
x
)
[
v
(
x
)
]
2
\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}
[
v
(
x
)
]
2
u
′
(
x
)
v
(
x
)
−
u
(
x
)
v
′
(
x
)
.
True
What rule is used to find the derivative of
tan
x
=
\tan x =
tan
x
=
sin
x
cos
x
\frac{\sin x}{\cos x}
c
o
s
x
s
i
n
x
?
Quotient rule
If
u
(
x
)
=
u(x) =
u
(
x
)
=
sin
x
\sin x
sin
x
, then
u
′
(
x
)
=
u'(x) =
u
′
(
x
)
=
cos
x
\cos x
cos
x
True
After applying the quotient rule to
sin
x
cos
x
\frac{\sin x}{\cos x}
c
o
s
x
s
i
n
x
, the expression simplifies to
1
cos
2
x
\frac{1}{\cos^{2} x}
c
o
s
2
x
1
, which is equal to \sec^{2} x.
Match the trigonometric function with its derivative:
sin
x
\sin x
sin
x
↔️
cos
x
\cos x
cos
x
cos
x
\cos x
cos
x
↔️
−
sin
x
- \sin x
−
sin
x
tan
x
\tan x
tan
x
↔️
sec
2
x
\sec^{2} x
sec
2
x
What is the derivative of
cos
x
\cos x
cos
x
?
−
sin
x
- \sin x
−
sin
x
Match the derivative rule with its formula:
Quotient rule ↔️
(
u
v
)
′
=
(\frac{u}{v})' =
(
v
u
)
′
=
u
′
v
−
u
v
′
v
2
\frac{u'v - uv'}{v^{2}}
v
2
u
′
v
−
u
v
′
Derivative of
sin
x
\sin x
sin
x
↔️
cos
x
\cos x
cos
x
Derivative of
cos
x
\cos x
cos
x
↔️
−
sin
x
- \sin x
−
sin
x
The quotient rule states that if
f
(
x
)
=
f(x) =
f
(
x
)
=
u
(
x
)
v
(
x
)
\frac{u(x)}{v(x)}
v
(
x
)
u
(
x
)
, then f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}</latex>.
True
Steps to find the derivative of
tan
x
\tan x
tan
x
using the quotient rule
1️⃣ Recognize \tan x = \frac{\sin x}{\cos x}</latex>
2️⃣ Identify
u
(
x
)
=
u(x) =
u
(
x
)
=
sin
x
\sin x
sin
x
and
v
(
x
)
=
v(x) =
v
(
x
)
=
cos
x
\cos x
cos
x
3️⃣ Find
u
′
(
x
)
=
u'(x) =
u
′
(
x
)
=
cos
x
\cos x
cos
x
and
v
′
(
x
)
=
v'(x) =
v
′
(
x
)
=
−
sin
x
- \sin x
−
sin
x
4️⃣ Apply the quotient rule
5️⃣ Use
sin
2
x
+
\sin^{2} x +
sin
2
x
+
cos
2
x
=
\cos^{2} x =
cos
2
x
=
1
1
1
to simplify
6️⃣ Conclude that
d
d
x
(
tan
x
)
=
\frac{d}{dx}(\tan x) =
d
x
d
(
tan
x
)
=
sec
2
x
\sec^{2} x
sec
2
x
Match the trigonometric identity with its formula:
Sine ↔️ \sin(x)</latex>
Cosine ↔️
cos
(
x
)
\cos(x)
cos
(
x
)
Tangent ↔️
tan
(
x
)
=
\tan(x) =
tan
(
x
)
=
sin
(
x
)
cos
(
x
)
\frac{\sin(x)}{\cos(x)}
c
o
s
(
x
)
s
i
n
(
x
)
What are the bounded limits of
sin
x
\sin x
sin
x
?
-1 and 1
The trigonometric identity for
tan
x
\tan x
tan
x
is \frac{\sin x}{\cos x}
What are the basic properties of trigonometric functions?
Periodicity, symmetry, boundedness
The derivative of
sin
x
\sin x
sin
x
is
cos
x
\cos x
cos
x
.
True
What is the quotient rule for differentiation?
d
d
x
(
u
v
)
=
\frac{d}{dx} \left(\frac{u}{v}\right) =
d
x
d
(
v
u
)
=
u
′
v
−
u
v
′
v
2
\frac{u'v - uv'}{v^{2}}
v
2
u
′
v
−
u
v
′
The derivative of
tan
x
\tan x
tan
x
is \sec^{2} x
The derivative of
cot
x
\cot x
cot
x
is - \csc^{2} x
The key trigonometric identities are used to simplify and manipulate trigonometric
expressions
What is one basic property of trigonometric functions related to their repetition after a certain interval?
Periodicity
The values of sine and cosine are always between
-1
and 1.
True
Match the trigonometric identity with its formula:
Tangent ↔️ `tan(x) = sin(x) / cos(x)`
Cotangent ↔️ `cot(x) = cos(x) / sin(x)`
Secant ↔️ `sec(x) = 1 / cos(x)`
Cosecant ↔️ `csc(x) = 1 / sin(x)`
Trigonometric functions are periodic, meaning they repeat their values at regular
intervals
.
True
What is the derivative of
cos
x
\cos x
cos
x
?
−
sin
x
- \sin x
−
sin
x
What is the simplified derivative of
tan
x
\tan x
tan
x
?
1
cos
2
x
\frac{1}{\cos^{2} x}
c
o
s
2
x
1
If f(x) = \frac{u(x)}{v(x)}</latex>, then
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
u
′
(
x
)
v
(
x
)
−
u
(
x
)
v
′
(
x
)
[
v
(
x
)
]
2
\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}
[
v
(
x
)
]
2
u
′
(
x
)
v
(
x
)
−
u
(
x
)
v
′
(
x
)
is the formula for the quotient rule.
If
v
(
x
)
=
v(x) =
v
(
x
)
=
cos
x
\cos x
cos
x
, what is
v
′
(
x
)
v'(x)
v
′
(
x
)
?
−
sin
x
- \sin x
−
sin
x
What is the derivative of
tan
x
\tan x
tan
x
?
sec
2
x
\sec^{2} x
sec
2
x
Steps to find the derivative of
tan
x
\tan x
tan
x
using the quotient rule:
1️⃣ Define
tan
x
\tan x
tan
x
as
sin
x
cos
x
\frac{\sin x}{\cos x}
c
o
s
x
s
i
n
x
2️⃣ Apply the quotient rule
3️⃣ Find the derivatives of
sin
x
\sin x
sin
x
and
cos
x
\cos x
cos
x
4️⃣ Substitute the derivatives into the quotient rule
5️⃣ Simplify the expression using trigonometric identities
The derivative of
sin
x
\sin x
sin
x
is
cos
x
\cos x
cos
x
True
The trigonometric identity
cos
2
x
+
\cos^{2} x +
cos
2
x
+
sin
2
x
=
\sin^{2} x =
sin
2
x
=
1
1
1
is used to simplify the expression for the derivative of
tan
x
\tan x
tan
x
to \sec^{2} x.
What is the derivative of
tan
x
\tan x
tan
x
?
sec
2
x
\sec^{2} x
sec
2
x
What is the simplified form of
1
cos
2
x
\frac{1}{\cos^{2} x}
c
o
s
2
x
1
?
sec
2
x
\sec^{2} x
sec
2
x
Which trigonometric identity is used to simplify the derivative of
cot
x
\cot x
cot
x
?
sin
2
x
+
\sin^{2} x +
sin
2
x
+
cos
2
x
=
\cos^{2} x =
cos
2
x
=
1
1
1
Trigonometric functions are
periodic
and repeat at regular intervals.
True
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