Trigonometric Identities for Double Angles
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Trigonometric identities
LO: To find the Pythagorean identity and trigonometric
identities for the sine and cosine of the double
angle.
To use those identities to solve problems.
28 September 2024
y
x
0
Pythagorean identity
From the unit circle we found that:
In this section we are going to look at special kinds of
equations called identities
B
1
OC
=
cos
Since it is a right angled triangle we can
use Pythagoras theorem and say that
A
C
cos
sin
BC
=
sin
cos
=
x
sin
=
y
(cos
sin
)
(cos
sin
)
2
=
1
cos
sin
=
1
cos
sin
tan
=
We can also say that
This called The
Pythagorean identity
Another identity
y
x
O
Double angle identity for cosine
In the following diagram, we draw the reflection of the
angle
in the fourth quadrant.
The length AC
The length AB can be found using the
cosine rule in the
AOB.
A
1
1
We can see that the angle AOB = 2
C
Now we have two expressions for AB
B
So, AB = AC + BC = 2
sin
AB
2
= AO
2
+ BO
2
– 2(AO)(BO)cos
(
2
AB
2
= 1
2
+ 1
2
– 2(1)(1)
cos
(
2
2
sin
Squaring both sides
4
sin
2
= 2 - 2
cos
(
2
Rearranging
2
cos
(
2
= 2 - 4
sin
2
Dividing by 2
cos
(
2
= 1 - 2
sin
2
sin
= BC
=
sin
sin
This is one identity for the cosine of the double angle
We will use the Pythagorean identity
We are going to use this identity to find more identities
for the cosine of the double angle
Rearranging the identity
cos
(
2
= 1 - 2
sin
2
cos
sin
=
1
sin
=
1 – cos
Substituting in the identity
cos
(
2
= 1 – 2
(
1 – cos
)
cos
(
2
= 1
–
2
+
2
cos
cos
(
2
= 2
cos
–
1
Expanding brackets
Simplifying and rearranging
Double angle identity for cosine
This is another identity for the cosine of the double angle
We will use the Pythagorean identity
We are going to use this last identity to find more
identities for the cosine of the double angle
We can substitute the identity
into the identity
cos
sin
=
1
cos
(
2
= 2
cos
–
(
cos
sin
)
Expanding brackets
cos
(
2
= 2
cos
–
cos
–
sin
cos
(
2
= 2
cos
–
1
Simplifying
cos
(
2
=
cos
– sin
Double angle identity for cosine
This is another identity for the cosine of the double angle
Now we are going to find a double-angle identity for sine
We can say
Equating these equations
Factorising
Double angle identity for sine
cos
(
2
= 1 - 2
sin
2
We will use the Pythagorean identity
cos
sin
=
1
cos
(2
sin
(2
=
1
Rearranging
cos
(2
=
1
– sin
(2
From the double angle identity for cosine
Squaring both sides
cos
(
2
= (1 - 2
sin
2
)
cos
(
2
= 1 - 4
sin
2
4
sin
4
1
– sin
(2
=
1
–
4
sin
2
4
sin
4
=
sin
(2
4
sin
2
–
4
sin
4
=
sin
(2
4
sin
2
(
1 – sin
2
)
Substituting from
=
sin
(2
4
sin
2
cos
2
Taking square root
of both sides
=
sin
(2
2
sin
cos
Simplifying
1 – sin
2
Double angle identities for cosine
Pythagorean identity
Double angle identity for sine
cos
sin
=
1
cos
(
2
=
2
cos
–
1
cos
(
2
=
cos
– sin
Summary of identities
cos
(
2
=
1
–
2
sin
2
sin
(
2
=
2
sin
cos
cos
sin
tan
=
Using the Pythagorean identity
Given that
Substituting the value of sine
cos
sin
=
1
Using identities
sin
, and 0
o
<
<
90
o
Find the exact values of
(a) cos
Rearranging and simplifying
Taking square root
of both sides
Solution:
Using the double angle identity for sine
Given that
Substituting the value of sin
and cos
Using identities
sin
, and 0
o
<
<
90
o
Find the exact values of
(b) sin
(2
)
Simplifying
Solution:
sin
(
2
=
2
sin
cos
sin
(
2
sin
(
2
sin
(
2
Using a double angle identity for cosine
Given that
Substituting the value of sin
Using identities
sin
, and 0
o
<
<
90
o
Find the exact values of
(c) cos
(2
)
Simplifying
Solution:
cos
(
2
=
1
–
2
sin
2
sin
(
2
Using identity of tangent
Given that
Changing
by 2
Using identities
sin
, and 0
o
<
<
90
o
Find the exact values of
(d) tan
(2
)
Substituting
sin
(
2
and
cos
(
2
Solution:
sin
(
2
cos
sin
tan
=
cos
(
2
)
sin
(
2
)
tan
(
2
)
=
tan
(
2
)
=
Simplifying
tan
(
2
)
=
Using the double angle identity for sine
Solve the equation
Replacing
sin
2
x
Using identities when working with equations
sin
2
x
sin
x
for 0
o
x
360
o
Do not use GDC
Rearranging
Solution:
Using the null factor law
sin
(
2
=
2
sin
cos
2(
sin
x
)(
cos
x
)
sin
x
2(
sin
x
)(
cos
x
)
– sin
x
Factorising
(
sin
x
)(2
cos
x
– 1
)
(
sin
x
)
(2
cos
x
– 1
)
or
If
sin
x
Then
x
If 2
cos
x
– 1
So
x
Then
x
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Special identities like the Pythagorean identity and double angle identities for sine and cosine are explored in this content. The Pythagorean identity states that cosine squared plus sine squared equals one, while the double angle identities provide formulas for cosine of double angles. Through the use of these identities, various relationships between trigonometric functions can be derived and utilized to solve problems effectively.
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Presentation Transcript
28 September 2024 Trigonometric identities LO: To find the Pythagorean identity and trigonometric identities for the sine and cosine of the double angle. To use those identities to solve problems. www.mathssupport.org www.mathssupport.org
Pythagorean identity In this section we are going to look at special kinds of equations called identities y From the unit circle we found that: OC = cos BC = sin sin = y cos = x (cos sin ) B 1 sin Since it is a right angled triangle we can use Pythagoras theorem and say that (cos ) + (sin )2 = 1 cos + sin = 1 This called The Pythagorean identity A C 0 x cos We can also say that sin Another identity tan = cos www.mathssupport.org www.mathssupport.org
Double angle identity for cosine In the following diagram, we draw the reflection of the angle in the fourth quadrant. The length AC A 1 We can see that the angle AOB = 2 sin y = BC = sin So, AB = AC + BC = 2sin sin C O x The length AB can be found using the cosine rule in the AOB. Squaring both sides 4sin2 = 2 - 2cos(2 ) Rearranging 2cos(2 ) = 2 - 4sin2 Dividing by 2 cos(2 ) = 1 - 2sin2 1 B AB2 = AO2 + BO2 2(AO)(BO)cos(2 ) AB2 = 12 + 12 2(1)(1)cos(2 ) AB = 2 2cos(2?) Now we have two expressions for AB 2sin = 2 2cos(2?) This is one identity for the cosine of the double angle www.mathssupport.org www.mathssupport.org
Double angle identity for cosine We are going to use this identity to find more identities for the cosine of the double angle cos(2 ) = 1 - 2sin2 We will use the Pythagorean identity cos + sin = 1 Rearranging the identity Substituting in the identity sin = 1 cos cos(2 ) = 1 2(1 cos ) Expanding brackets Simplifying and rearranging This is another identity for the cosine of the double angle cos(2 ) = 1 2 + 2cos cos(2 ) = 2cos 1 www.mathssupport.org www.mathssupport.org
Double angle identity for cosine We are going to use this last identity to find more identities for the cosine of the double angle cos(2 ) = 2cos 1 We will use the Pythagorean identity cos + sin = 1 We can substitute the identity into the identity cos(2 ) = 2cos (cos + sin ) Expanding brackets cos(2 ) = 2cos cos sin Simplifying cos(2 ) = cos sin This is another identity for the cosine of the double angle www.mathssupport.org www.mathssupport.org
Double angle identity for sine Now we are going to find a double-angle identity for sine We will use the Pythagorean identity cos (2 )+ sin (2 ) = 1 Rearranging cos (2 )= 1 sin (2 ) From the double angle identity for cosine cos + sin = 1 We can say cos(2 ) = 1 - 2sin2 cos (2 ) = (1 - 2sin2 ) cos (2 ) = 1 - 4sin2 + 4sin4 1 sin (2 )= 1 4sin2 + 4sin4 =sin (2 ) 4sin2 4sin4 =sin (2 ) 4sin2 (1 sin2 ) =sin (2 ) 4sin2 cos2 2sin cos Squaring both sides Equating these equations Simplifying Factorising Substituting from Taking square root of both sides 1 sin2 =sin(2 ) www.mathssupport.org www.mathssupport.org
Summary of identities sin tan = cos Pythagorean identity cos + sin = 1 Double angle identities for cosine cos(2 ) = 1 2sin2 cos(2 ) = 2cos 1 cos(2 ) =cos sin Double angle identity for sine sin(2 ) = 2sin cos www.mathssupport.org www.mathssupport.org
Using identities sin =3 4, and 0o< < 90o Given that Find the exact values of (a) cos Solution: Using the Pythagorean identity cos + sin = 1 Substituting the value of sine 2= 1 3 4 cos + Rearranging and simplifying 9 cos = 1 16 cos = 7 16 Taking square root of both sides 7 cos = 4 www.mathssupport.org www.mathssupport.org
7 Using identities cos = 4 sin =3 4, and 0o< < 90o (b) sin(2 ) Given that Find the exact values of Solution: Using the double angle identity for sine sin(2 )= 2sin cos Substituting the value of sin and cos sin(2 ) = 2 3 4 7 4 Simplifying 6 7 16 sin(2 ) = sin(2 ) = 3 7 8 www.mathssupport.org www.mathssupport.org
7 Using identities cos = 4 sin =3 sin(2 )= 3 7 4, and 0o< < 90o (c) cos(2 ) Given that 8 Find the exact values of Solution: Using a double angle identity for cosine cos(2 ) = 1 2sin2 Substituting the value of sin 2 3 4 cos(2 ) = 1 2 Simplifying 9 cos(2 ) = 1 2 16 9 8 cos(2 ) = 1 cos(2 ) = 1 8 www.mathssupport.org www.mathssupport.org
7 Using identities cos = 4 sin =3 sin(2 )= 3 7 cos(2 ) = 1 4, and 0o< < 90o (d) tan(2 ) Given that 8 Find the exact values of Solution: 8 Using identity of tangent sin tan = cos sin(2 ) Changing by 2 tan(2 )= 3 7 8 cos(2 ) tan(2 )= Substituting sin(2 )and cos(2 ) 1 8 tan(2 )=3 7 Simplifying 8 8 1 tan(2 )= 3 7 www.mathssupport.org www.mathssupport.org
Using identities when working with equations Solve the equation Do not use GDC Solution: for 0o x 360o sin2x = sin x Using the double angle identity for sine sin(2 )= 2sin cos 2(sin x)(cos x)= sin x 2(sin x)(cos x) sin x = (sin x)(2cos x 1)= (sin x) = If sin x = Then x = If 2cos x 1 = Then cos x =1 Then x = Replacing sin 2x Rearranging Factorising or Using the null factor law (2cos x 1)= 2So x = www.mathssupport.org www.mathssupport.org
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org www.mathssupport.org www.mathssupport.org
