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

3.TRIGONOMETRY

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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

3.TRIGONOMETRY

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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

3.TRIGONOMETRY

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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