Download Verify a Trigonometric Identity

Five-Minute Check (over Lesson 5-1)
Then/Now
New Vocabulary
Example 1: Verify a Trigonometric Identity
Example 2: Verify a Trigonometric Identity by Combining
Fractions
Example 3: Verify a Trigonometric Identity by Multiplying
Example 4: Verify a Trigonometric Identity by Factoring
Example 5: Verify an Identity by Working Each Side Separately
Concept Summary: Strategies for Verifying Trigonometric
Identities
Example 6: Determine Whether an Equation is an Identity
Over Lesson 5-1
Find the value of the expression using the given
information.
If tan θ =
A.
B.
C.
, find cot θ.
Over Lesson 5-1
Find the value of the expression using the given
information.
If sin θ =
A.
B.
C.
and cos θ =
, find tan θ.
Over Lesson 5-1
Find the value of the expression using the given
information.
If csc θ = 3 and cos θ < 0, find cos θ and tan θ.
A.
B.
C.
Over Lesson 5-1
Simplify csc x – csc x cos 2 x.
A. sin x
B. cos x
C. csc x(1 + sin x)
D. 1 – cos x
Over Lesson 5-1
If sin θ = 0.59, find
A. −0.81
B. −0.59
C. 0.59
D. 0.81
.
You simplified trigonometric expressions. (Lesson 5-1)
• Verify trigonometric identities.
• Determine whether equations are identities.
• verify an identity
Verify a Trigonometric Identity
Verify that
.
The left-hand side of this identity is more complicated,
so transform that expression into the one on the right.
Pythagorean Identity
Reciprocal Identity

Simplify.
Verify a Trigonometric Identity
Answer:
Verify that 2 – cos2 x = 1 + sin2 x.
A. 2 – cos2x = –(sin2x + 1) + 2 = 1 + sin2x
B. 2 – cos2x = 2 – (sin2x + 1) = 1 + sin2x
C. 2 – cos2x = 2 – (1 + sin2x) + 2 = 1 + sin2x
D. 2 – cos2x = 2 – (1 – sin2x) = 1 + sin2x
Verify a Trigonometric Identity by Combining
Fractions
Verify that
.
The right-hand side of the identity is more
complicated, so start there, rewriting each fraction
using the common denominator 1 – cos2 x.
Start with the
right hand side
of the identity.
Common
denominator
Distributive
Property
Verify a Trigonometric Identity by Combining
Fractions
Simplify.
Divide out the
common factor
of sin x.
Simplify.

Quotient Identity
Verify a Trigonometric Identity by Combining
Fractions
Answer:
Verify that
A.
B.
C.
D.
.
Verify a Trigonometric Identity by Multiplying
Verify that
.
Because the left-hand side of this identity involves a
fraction, it is slightly more complicated than the right
side. So, start with the left side.
Multiply the
numerator and
denominator by
the conjugate of
sec x – 1, which
is sec x + 1.
Multiply.
Verify a Trigonometric Identity by Multiplying
Pythagorean
Identity
Quotient Identity
Multiply by the
reciprocal of the
denominator.
Divide out the
common factor
of sin x.
Verify a Trigonometric Identity by Multiplying
Distributive
Property
Rewrite the
fraction as the
sum of two
fractions;
Reciprocal
Identity.
Divide out the
common factor
of cos x.
Verify a Trigonometric Identity by Multiplying

Quotient Identity
Verify a Trigonometric Identity by Multiplying
Answer:
Verify that
A.
B.
C.
D.
.
Verify a Trigonometric Identity by Factoring
Verify that cos x sec 2 x tan x – cos x tan3 x = sin x.
cos x sec x tan x – cos x tan x
= cos x tan x (sec2 x – tan2 x)
Start with the lefthand side of the
identity. Factor.
= cos x tan x (1)
Pythagorean Identity
=
Quotient Identity
= sin x 
Divide out the
common factor of
cos x.
2
3
Verify a Trigonometric Identity by Factoring
Answer: cos x sec 2 x tan x – cos x tan3 x = cos x tan
x (sec2 x – tan2 x) = cos x tan x (1)
=
= sin x
Verify that
csc x – cos x csc x – cos x cot x + cot x = sin x.
A.
B.
C.
D.
Verify an Identity by Working Each Side
Separately
Verify that
.
Both sides look complicated, but there is a clear first
step for the expression on the left. So, start with the
expression on the left.
Write as the sum of two
fractions.
Simplify and apply a
Reciprocal Identity.
Verify an Identity by Working Each Side
Separately
From here, it is unclear how to transform 1 + cot x into
, so start with the right side and work to
transform it into the intermediate form 1 + cot x.
Pythagorean Identity
Simplify.
Factor.
Verify an Identity by Working Each Side
Separately
Divide out the
common factor of
1 – cot x.
To complete the proof, work backward to connect the
two parts of the proof.
Write as the sum of
two fractions.
Simplify and apply a
Reciprocal Identity.
Multiply by
.
Verify an Identity by Working Each Side
Separately
Simplify.
Pythagorean Identity

Answer:
Simplify.
Verify that tan2 x – sin2 x = sin2 x tan2 x.
A.
B.
C.
D.
Determine Whether an Equation is an Identity
A. Use a graphing calculator to test whether
is an identity. If it appears to be
an identity, verify it. If not, find an x-value for
which both sides are defined but not equal.
The equation appears to be an identity because the
graphs of the related functions over [–2π, 2π] scl: π
by [–1, 3] scl: 1 coincide. Verify this algebraically.
Determine Whether an Equation is an Identity
Pythagorean Identity
Divide out the common
factor of sec x.
Determine Whether an Equation is an Identity
Reciprocal Identities
Simplify.
Quotient Identity
Answer:
Determine Whether an Equation is an Identity
B. Use a graphing calculator to test whether
is an identity. If it
appears to be an identity, verify it. If not, find an
x-value for which both sides are defined but not
equal.
Determine Whether an Equation is an Identity
The graphs of the related functions do not coincide for
all values of x for which the both functions are defined.
When
, Y1 ≈ 1.43 but Y2 ≈ –0.5. The equation is
not an identity.
Determine Whether an Equation is an Identity
Answer: When
, Y1 ≈ 1.43 but Y2 = –0.5. The
equation is not an identity.
Use a graphing calculator to test whether
is an identity. If it appears
to be an identity, verify it. If not, find a value for
which both sides are defined but not equal.
A. The equation appears to be an identity because
the graphs of the related functions over
[–2π, 2π] scl: π by [–3, 3] scl: 1 coincide.
B. When
, Y1 ≈ 0.71 but Y2 ≈ 0.29. The
equation is not an identity.