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Lesson 6
Trigonometric Function (III)
6A • Trigonometric Identities • Exact Values of Some Trigonometric Function
Course I
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Sum and Difference Identities for Sine and Cosine
Sum and Difference Formulae
βαβαβαβαβαβαβαβαβαβαβαβα
sinsincoscos)cos(sinsincoscos)cos(sincoscossin)sin(sincoscossin)sin(
+=−−=+−=−+=+
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.
This formula is most important because all the identities in the following are based on this formula.
Memorize !
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Proof of Sum Formula for Sine
)sin( βα +DE = AD =FG + DH =FA +DF =AD + AD
αsinαcosαβ sincos βsin
αcos
βαβαβα sincoscossin)sin( +=+
Therefore
Example 1 Prove the formula βαβαβα sincoscossin)sin( +=+
We can prove these identities geometrically.
From the figure
Ans.
4
Find Exact Values of Trigonometric Function We can find values of some trigonometric functions from known cases.
2 1
1
12
3
Example 2 Find the values of (a) sin15°and (b) cos15°. Ans.
426
2213
21
21
23
21
sin30sin45cos30cos45)3045cos(15cos
,4
262213
21
21
23
21
30sin45cos30cos45sin)3045sin(15sin
+=
+=×+×=
°°+°°=°−°=°
−=
−=×−×=
°°−°°=°−°=°
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Sum and Difference Identities for Tangent Example 3 Derive the sum and difference formulae of the tangent function.
βαβα
βαβαββ
αα
βαβαβαβα
βαβα
βαtantan1tantan
coscossinsin1
cossin
cossin
sinsincoscossincoscossin
)cos()sin()tan(
−
+=
−
+=
−
+=
+
+=+
βαβα
βαβαββ
αα
βαβαβαβα
βαβα
βαtantan1tantan
coscossinsin1
cossin
cossin
sinsincoscossincoscossin
)cos()sin()tan(
+
−=
+
−=
+
−=
−
−=−
βα coscos÷
In summary
βαβα
βαβαβα
βαtantan1tantan)tan(,
tantan1tantan)tan(
+
−=−
−
+=+
We use the sine and cosine formulae.
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Exercise
Pause the video and solve the problem.
Ans. Exercise 1 Find the values of (a) sin75°, (b) cos75°and (c) tan75°.
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Answer to the Exercise
Ans. Exercise 1 Find the values of (a) sin75°, (b) cos75°and (c) tan75°.
426
2213
21
21
23
21
30sin45cos30cos45sin)3045sin(75sin
+=
+=×+×=
°°+°°=°+°=°
426
2213
21
21
23
21
30sin45sin30cos45cos )3045cos(75cos−
=−
=×−×=
°°−°°=°+°=°
1313
3/1113/11
30tan45tan130tan45tan )3045tan(75tan
−
+=
×−
+=
°°−°+°
=°+°=°
(a) (b)
(c)
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Exercise
Ans.
Exercise 2 There are two lines and 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. (2) Find the angle θ (0<θ< ) made by these two lines.
12113 +=−= xyxy
2π
Pause the video and solve the problem.
Uhh….
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Exercise
Ans.
Exercise 2 There are two lines and 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. (2) Find the angle θ (0<θ< ) made by these two lines.
12113 +=−= xyxy
2π
21tan,3tan)1( == βα
βαθ −=)2(
4,1
2131213
tantan1tantan)tan(tan π
θβαβα
βαθ =∴=×+
−=
+
−=−=
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Lesson 6
Trigonometric Function (II)
6B6B
• Derivation of Trigonometric Identities • Double-Angle Identities • Half-Angle Identities • Product-to-Some Identities • Some-to-Product Identities
Course I
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Derivation of Trigonometric Identities
βα =
Double-Angle Identities etc.
Half-Angle Identities etc.
αβ 2=
Three-Angle Identities etc.
Product-to-Sum Identities etc.
Sum-to-Product Identities etc.
Solve inversely Put
2α
α →
βαβαβαβαβαβαsinsincoscos)cos(
sincoscossin)sin(∓=±
±=±
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Double-Angle Identities
Example 4 Prove the double angle identity of tangent
Ans.
αα
αααα
αααα
αα
α
2
22
tan1tan2
sincoscossin2
)cos()sin(
2cos2sin2tan
−=
−=
+
+=
= : From definition
: From sum identities
: Divide by and use
α2cos
αα
αcossintan =
1cos2sin21sincoscos2
cossin2sin2
22
22
−=−=
−=
=
αα
ααα
ααα
αα
α 2tan1tan22tan
−=
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Half-Angle Identities
Example 5 Evaluate the values of
Ans. Since 22.5°is in quadrant I, all these values are positive.
Therefore
2225.22cos
422
245cos15.22cos
2225.22sin
422
22122/
211
245cos15.22sin
2
2
+=∴
+=
+=
−=∴
−=
−=⎟
⎠
⎞⎜⎝
⎛−=
−=
!!
!
!!
!
12222
24)22(
22
225.22cos5.22sin5.22tan
2
−=−
=−−
=+
−== !
!!
2cos1
2cos 2 αα +
=ααα
cos1cos1
2tan2
+
−=2
cos12
sin 2 αα −=
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Product-to-Sum Identities
Example 6 Evaluate the values of
Ans.
)}cos(){cos(21sinsin
)}cos(){cos(21coscos
)}sin(){sin(21sincos
)}sin(){sin(21cossin
βαβαβα
βαβαβα
βαβαβα
βαβαβα
−−+−=
−++=
−−+=
−++=
41cos60
21)}60cos({cos90
21
)}7515cos()7515{cos(2175sin15sin
=°=°−−°−=
°−°−°+°−=°°
°° 75sin15sin
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Sum-to-Product Identities
Example 7 Evaluate the values of Ans.
P-to-S
S-to-P
BA
=−=+
βαβα
2
sin2
sin2coscos
2cos
2cos2coscos
2sin
2cos2sinsin
2cos
2sin2sinsin
BABABA
BABABA
BABABA
BABABA
−+−=−
−+=+
−+=−
−+=+
°+°+° cos230cos110cos10
0cos110cos110 cos110)110cos(2cos120
cos110)cos230(cos10cos230cos110cos10
=°+°−=°+°−°=
°+°+°=°+°+°
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Exercise
Ans.
Pause the video and solve the problem.
Exercise 3 Let and . Express the following functions by and . (1) (2)
θθ cossin == cscs
θθ 4cos4sin
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Answer to the Exercise Exercise 3 Let and . Express the following functions by and . (1) (2)
θθ cossin == cscs
θθ 4cos4sin
Ans.
18c8c11)4c2(4c 1)12(2
11)2(2cos122cos)2cos(2cos4 (2)
)2ssc(14cos2 2sin2)2sin(2sin4 )1(
2424
22
222
2
+−=−+−=
−−=
−−=−=×=
−==×=
cθθθθ
θθθθ
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Exercise
Ans.
1cossin 22 =+ xxExercise 4 When , evaluate [Hint] Divide the expression by .
2tan −=x xx 2cos2sin +
Pause the video and solve the problem by yourself.
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Answer to the Exercise
Ans.
1cossin 22 =+ xxExercise 4 When , evaluate [Hint] Divide the expression by .
2tan −=x xx 2cos2sin +
57
12)(2)(12)2(
1tantan12tan
cossinsincoscos2sin
12cos2sin2cos2sin
2
2
2
2
22
22
−=+−−−+−
=+−+
=
+−+
=+
=+
xxx
xxxxxxxxxx