Solve each quadratic equation by completing the square. In ...
TRIGONOMETRY NOTES (2013). Simple Identities! IDENTITIES DIVIDE BY RESULT! Example 2: "1. Prove the...
Transcript of TRIGONOMETRY NOTES (2013). Simple Identities! IDENTITIES DIVIDE BY RESULT! Example 2: "1. Prove the...
3.1. Three More Trigonometric Functions
Youʼve learnt from P1, the following Trigonometric Functions:
•••
What happen if we inverted these trigonometric function?
Example 1 :
Solve the following equations for x between 0 and 360.
(a)" " " " (b) " " " " (c)
(d) " " " " (e) " " " " (f)
3.TRIGONOMETRY
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! secant
! cosecant
! cotangent
cosec x = 2 3 cotx = 4 secx = 5 sin 20
cot2 x = 3 sinx(cosec x−√2) = 0 cotx (cos2 x− 4) = 0
sin θcos θtan θ
sec θ =1
cos θ, cos θ �= 0
cosec θ =1
sin θ, sin θ �= 0
cot θ =1
tan θ, tan θ �= 0
3.2. Graphs of SEC , COSEC and COT
(a) SIN graph COSEC graph
(b) COS graph" SEC graph
(c) TAN graph" COT graph
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3.3. Simple Identities
!IDENTITIES DIVIDE BY RESULT
! Example 2:
" 1. Prove the following identities:
" (a)" (b)
" 2. Solve the equation for " 3. Solve the equation for
3.TRIGONOMETRY
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tan θ =sin θ
cos θ, cos θ �= 0
sin2 θ + cos2 θ = 1
sin2 θ + cos2 θ = 1
sin2 θ + cos2 θ = 1
sin2 θ
cos2 θ
sec θ − cos θ ≡ sin θ tan θ sec2 θ + cosec2 θ ≡ sec2 θ cosec2 θ
sec2 θ + tan θ − 1 = 0sec θ = 3 cos θ + sin θ
0◦ ≤ θ ≤ 360◦ .0◦ ≤ θ ≤ 360◦ .
3.4. Compound Angle Formulae
COMPOUND ANGLE
! Recall:
0 30 45 60 90
Sin
Cos
Tan
If
If
If
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sin (A+B)
sin (A−B)
cos (A−B)
tan (A−B)
cos (A+B)
tan (A+B)
sin θ = 0
cos θ = 0
tan θ = 0
! Example 3:
" 1. Simplify the following expressions:
" (a) " (b)
" (c)" (d)
" 2. Prove the following identities:
" (a)
" (b)
" (c)
" 3. Find all the angles between 0 and 360 which satisfy the following equations:"" (a) " (b)
" (c)" (d)
3.TRIGONOMETRY
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sin 2θ cos θ − cos 2θ sin θ cos 3θ cos 2θ + sin 3θ sin 2θ
sin θ cos 2θ − cos θ sin 2θ tan 2θ − tan 5θ
1 + tan 2θ tan 5θ
cos(A+B)− cos(A−B) ≡ −2 sinA sinB
sin(A+B)
cos(A−B)≡ tanA+ tanB
tanA− tanB
tanA+ tanB ≡ sin(A+B)
cosA cosB
sin(x+ 30) = 2 cosx 3 sinx = 2 cos(x+ 45)
2 tanx+ 3 tan(x− 45) = 0 2 sec(x+ 60) = 5 sec(x− 20)
3.5. Double Angle Formulae
! (i)
" Let
" (ii)
" Let
"
" (iii)
" " Let
"
3.TRIGONOMETRY
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sin(A+B) = sinA cosB + cosA sinB
cos(A+B) = cosA cosB − sinA sinB
tan(A+B) =tanA+ tanB
1− tanA tanB
A = θ , B = θ
A = θ , B = θ
A = θ , B = θ
" Important Summary;
(i)
(ii)
(iii)
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sin 2θ
cos 2θ
tan 2θ
sin 4θ =
sin θ =
cos 4θ =
tan θ =
" Example 4 :
" 1. Prove the following identities:
" (i)" (ii)
" (iii)" (iv)
" 2. Find the angles between 0 and 360"" (i)" (ii)
3.TRIGONOMETRY
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(cosx− sinx)2 ≡ 1− 2 sinx tanx+ cotx ≡ 2 cosec 2x
cosec2A+ cot 2A ≡ cotAtan 2A− 2 tan 2A sin2 A ≡ sin 2A
4 sin 2x = sinx tan 2x tanx = 3
3.6. The Expression:
" where (always ____________ ) and " " (always ___________ )
"
" Example 5:
" 1. Convert in the form " where R is positive and is acute.
" 2. Express into where ! and
" 3. (a) Solve where
" (b) Solve where
" 4. (a) Find the maximum and minimum of " and the values of " which give the
" " maximum and minimum and the corresponding values of for
" (b) Find the maximum and minimum values of " " and the values of which
" "
" " give the maximum and minimum and the corresponding values of for
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GENERAL FORMULA:
(1)
(2)
where ! ! and
a cos θ ± b sin θ =
a sin θ ± b cos θ =
R > 0 0 < α < 90
a cos θ ± b sin θ ≡ R cos(θ ∓ α)
a sin θ ± b cos θ ≡ R sin(θ ± α)
R =�a2 + b2 α = tan−1
�b
a
�
3 cos θ + 4 sin θ R cos (θ − α) α
2√2 = cos θ R sin (θ − α) R > 0 0◦ < α < 90◦ .
3 cosx+ 4 sinx = 2 −180◦ < x < 180◦ .
−180◦ < x < 180◦ .5 cos 2θ − 12 sin 2θ = 8
3 cos θ + 4 sin θ θ
θ 0◦ ≤ θ ≤ 360◦ .
0◦ ≤ θ ≤ 360◦ .θ
θ12
3 cos θ + 4 sin θ + 7