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Course I
Lesson 6
Trigonometric Function (III)　6A
•  Trigonometric Identities
•  Exact Values of Some Trigonometric Function
1
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Sum and Difference Identities for Sine and Cosine
An identity is an equality that is true for any value of the variable.
(An equation is an equality that is true only for certain values of the variable.)
In calculation, we may replace either member of the identity with the other.
We use an identity to give an expression a more convenient form.
Sum and Difference Formulae
sin(α + β ) = sinαcosβ + cosαsinβ
sin(α − β ) = sinαcosβ − cosαsinβ
cos(α + β ) = cosαcosβ − sinαsinβ
cos(α − β ) = cosαcosβ + sinαsinβ
Memorize !
This formula is most important because all the identities in the following
are based on this formula.
2
Proof of Sum Formula for Sine
We can prove these identities geometrically.
Example 1 Prove the formula sin(α + β ) = sinαcosβ + cosαsinβ
Ans.
From the figure
DE = AD sin(α + β )
=FG + DH
=FA sin α +DF cos α
=AD cos β sin α + AD sin β cos α
Therefore
sin(α + β ) = sin α cos β + cos α sin β
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Find Exact Values of Trigonometric Function
We can find values of some trigonometric functions from known cases.
2
1
1
2
1
3
Example 2 Find the values of (a) sin15°and (b) cos15°.
Ans.
sin 15° = sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°
1
3 1 1
3 −1
6− 2
×
−
× =
=
,
4
2 2
2 2
2 2
cos15° = cos(45° − 30°) = cos45°cos30° + sin45°sin30°
=
=
1
3
1 1
3 +1
6+ 2
×
+
× =
=
4
2 2
2 2
2 2
4
Sum and Difference Identities for Tangent
Example 3 Derive the sum and difference formulae of the tangent function.
We use the sine and cosine formulae.
÷ cosα cos β
sin α sin β
+
sin(α + β ) sin α cos β + cos α sin β cos α cos β
tan α + tan β
tan(α + β ) =
=
=
=
cos(α + β ) cos α cos β − sin α sin β 1 − sin α sin β 1 − tan α tan β
cos α cos β
sin α sin β
−
sin(α − β ) sin α cos β − cos α sin β cos α cos β
tan α − tan β
tan(α − β ) =
=
=
=
cos(α − β ) cos α cos β + sin α sin β 1 + sin α sin β 1 + tan α tan β
cos α cos β
In summary
tan(α + β ) =
tan α + tan β
,
1 − tan α tan β
tan(α − β ) =
tan α − tan β
1 + tan α tan β
5
Exercise
Exercise 1 Find the values of (a) sin75°, (b) cos75°and (c) tan75°.
Ans.
Pause the video and solve the problem.
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Answer to the Exercise
Exercise 1 Find the values of (a) sin75°, (b) cos75°and (c) tan75°.
Ans.
(a) sin 75° = sin( 45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
=
1
2
×
3
1 1
3 +1
6+ 2
+
× =
=
2
4
2 2
2 2
(b) cos 75° = cos( 45° + 30°) = cos 45° cos 30° − sin 45° sin 30°
=
1
3
1 1
3 −1
6− 2
×
−
× =
=
4
2 2
2 2
2 2
tan 45° + tan 30°
1 − tan 45° tan 30°
1 + 1/ 3
3 +1
=
=
1 − 1× 1/ 3
3 −1
(c) tan 75° = tan( 45° + 30°) =
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Exercise
1
Exercise 2 There are two lines y = 3x − 1 and y = x + 1
2
Answer the following questions.
(1)  Let the angles made by these lines and the x-axis
by α and β. Express the slopes of these lines by
tangent function values. π
(2)  Find the angle θ (0<θ< ) made by these two
2
lines.
Ans.
Pause the video and solve the problem.
Uhh….
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Exercise
1
Exercise 2 There are two lines y = 3x − 1 and y = x + 1
2
Answer the following questions.
(1)  Let the angles made by these lines and the x-axis
by α and β. Express the slopes of these lines by
tangent function values. π
(2)  Find the angle θ (0<θ< ) made by these two
2
lines.
Ans.
(1)
tan α = 3,
1
tan β =
2
(2) θ = α − β
1
3−
tan α − tan β
2 = 1,
tan θ = tan(α − β ) =
=
1 + tan α tan β 1 + 3 × 1
2
∴
θ=
π
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Course I
Lesson 6
Trigonometric Function (II)　６B6B
•  Derivation of Trigonometric Identities
•  Double-Angle Identities
•  Half-Angle Identities
•  Product-to-Some Identities
•  Some-to-Product Identities
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Derivation of Trigonometric Identities
Three-Angle Identities
Double-Angle Identities
etc.
α→
α
2
etc.
β = 2α
Half-Angle Identities
etc.
α =β
sin(α ± β ) = sinαcosβ ± cosαsinβ
cos(α ± β ) = cos α cos β ∓ sin α sin β
Solve inversely
Put
Product-to-Sum Identities
Sum-to-Product Identities
etc.
etc.
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Double-Angle Identities
sin2α = 2sinαcosα
cos2α = cos2α − sin 2α
= 1 − 2sin 2α = 2cos2α − 1
tan 2α =
2 tan α
1 − tan 2 α
Example 4 Prove the double angle identity of tangent
Ans.
sin 2α
cos 2α
sin(α + α )
2 sin α cos α
=
=
cos(α + α ) cos 2 α − sin 2 α
2 tan α
=
1 − tan 2 α
tan 2α =
: From definition
: From sum identities
2
: Divide by cos α
and use tan α = sin α
cos α
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Half-Angle Identities
α
sin 2
2
=
1 − cosα
2
1 + cosα
cos
=
2
2
2
α
1 − cosα
tan
=
2 1 + cosα
2
α
Example 5 Evaluate the values of
Ans.
Since 22.5°is in quadrant I, all these values are positive.
1 − cos 45! ⎛
1 ⎞
2 −1 2 − 2
sin 22.5 =
= ⎜1 −
=
⎟/2 =
2
4
2⎠
2 2
⎝
2− 2
∴ sin 22.5 =
2
1 + cos 45! 2 + 2
cos 22.5 =
=
2
4
∴ cos 22.5! =
2
2
!
!
!
2+ 2
2
Therefore
!
2
sin
22
.
5
2
−
2
(
2
−
2
)
2− 2
tan 22.5! =
=
=
=
= 2 −1
!
cos 22.5
4−2
2
2+ 2
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Product-to-Sum Identities
1
{sin(α + β ) + sin(α − β )}
2
1
cosα sin β = {sin(α + β ) − sin(α − β )}
2
1
cosα cos β = {cos(α + β ) + cos(α − β )}
2
1
sin α sin β = − {cos(α + β ) − cos(α − β )}
2
sin α cos β =
Example 6 Evaluate the values of
Ans.
sin 15° sin 75°
1
sin 15° sin 75° = − {cos(15° + 75°) − cos(15° − 75°)}
2
1
1
1
= − {cos90° − cos(−60°)} = cos60° =
2
2
4
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Sum-to-Product Identities
P-to-S
α +β = A
α −β =B
S-to-P
A+ B
A− B
cos
2
2
A+ B
A− B
sin A − sin B = 2 cos
sin
2
2
A+ B
A− B
cos A + cos B = 2 cos
cos
2
2
A+ B
A− B
cos A − cos B = −2 sin
sin
2
2
sin A + sin B = 2 sin
Example 7 Evaluate the values of
cos10° + cos110° + cos230°
Ans.
cos10° + cos110° + cos230° = (cos10° + cos230°) + cos110°
= 2cos120°cos( −110°) + cos110°
= −cos110° + cos110° = 0
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Exercise
c = cosθ
Exercise 3　Let　s = sin θ
and　．　Express
the following functions
by　s andc .
(1) sin 4θ
(2)　cos 4θ
Ans.
Pause the video and solve the problem.
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Answer to the Exercise
c = cosθ
Exercise 3　Let　s = sin θ
and　．　Express
the following functions
by　s andc .
(1) sin 4θ
(2)　cos 4θ
Ans.
(1) sin4θ = sin(2 × 2θ ) = 2sin2θ cos2θ = 4sc(1 − 2s 2 )
(2) cos4θ = cos(2 × 2θ ) = 2cos 2 2θ − 1 = 2(2cos 2θ − 1) 2 − 1
= 2(2c 2 − 1) 2 − 1
= 2(4c 4 − 4c 2 + 1) − 1 = 8c 4 − 8c 2 + 1
17
Exercise
Exercise 4 When tanx = −2 , evaluate sin 2 x + cos 2 x
[Hint] Divide the expression by sin 2 x + cos2 x = 1
.
Ans.
Pause the video and solve the problem by yourself.
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Answer to the Exercise
Exercise 4 When tanx = −2 , evaluate sin 2 x + cos 2 x
[Hint] Divide the expression by sin 2 x + cos2 x = 1
.
Ans.
sin 2 x + cos 2 x 2sin x cos x + cos2 x − sin 2 x
sin 2 x + cos 2 x =
=
1
sin 2 x + cos2 x
2tanx + 1 − tan 2 x 2( −2) + 1 − (−2) 2
7
=
=
=
−
tan 2 x + 1
(−2) 2 + 1
5
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