Mathematics Extension 1 Notes

7/27/2019 Mathematics Extension 1 Notes

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Chapter 6 Trigonometry

6.1: Trigonometric Ratios

1sin = ------

cosec

1 cos

cos = ------ cot = ------sec sin

1 sintan = ------ tan = ------

cot cos

6.2: Fundamental Trigonometric Identities

sin + cos = 1

- Substitute sin / cos into unit circle equation

- Dividing through by sin

sin cos 1------- + ------- = ------- 1 + cot = cosecsin sin sin

- Dividing through by cos

sin cos 1------- + ------- = ------- 1 + tan = seccos cos cos

6.3: The Cosine Rule

b + c - acos A = --------------

2bc

a = b + c - (2bc.cos A)

6.4: The Sine Rule

b c

a

When solvingtrigonometric proofs,

always work from one sideto the other until it equals

Given 3 sides,asked for an

Given 2 sides / 1angle, asked for a

side

?

b ?

c

A


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6.5: Sum and Difference of Angles (Extension 1)

~ sin (A + B) = sinA . cosB + cosA . sinB

~ sin (A B) = sinA . cosB cosA . sinB

e.g: find the exact value of sin75

sin75 = sin (45 + 30)

= (sin45 . cos30) + (sin30 . sin45)

1 x 3 1 x 1 1 + 3= -------- + -------- = --------

2 x 2 2 x 2 22

~ cos (A + B) = cosA . cosB - sinA . sinB

~ cos (A B) = cosA . cosB + sinA . sinB

e.g: find the exact value of cos105

cos105 = cos (60 + 45)

= (cos60 . cos45) + (sin60 . sin45)

1 x 1 3 x 2 1 + 6= -------- + -------- = ---------

2 x 2 2 x 1 2 + 22

~ tan (A + B) = tanA + tanB-------------------1 tanA . tanB

~ tan (A - B) = tanA - tanB

-------------------1 + tanA . tanB

e.g: expand and simplify tan ( + 45) 1

tan ( + 45) 1 = tan + tan45 tan + 1-------------------- - 1 = ---------- - 11 - tan + tan45 1 tan

tan + 1 (1 tan) 2tan= ------------------------ = ----------

1 tan 1 tan


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6.6: Ratios of Double Angles (Extension 1)

Using the sum and difference of angles:

sin (A + B) = sinA . cosB + sinB . cosA

- now, let B = A

:. sin (A + A) = sinA . cosA + cosA . sinA

sin2A = 2 (sinA . cosA)

cos (A + B) = cosA . cosB - sinA . sinB

- now, let B = A

:. cos (A + A) = cosA . cosA - sinA . sinA

cos2A = cosA - sinA

also, by substituting (1 - sinA) for cosA

:. cos2A = 1 - sinA - sinA

= 1 2sinA

Also, by substituting (1 - cosA) for sinA

:. cos2A = cosA (1 - cosA)

= cosA + cosA 1

= 2cosA 1

tan (A + B) = tanA + tanB------------------- tanA 11 tanA . tanB

- now, let B = A

:. tan (A + A) = tanA + tanA------------------1 tanA . tanA

tan2A = 2tanA-----------1 - tanA


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6.7: The t = tan ( ) formula (Half-Angle Formula)

Let t = tan ( )

Using the Double Angle Formula

sin2 = 2 (sin . cos)Divide through by 2

:. sin = 2 (sin cos ) sub in t for

1 1 2t= 2 . ---------- . ---------- :. sin = --------

(1 + t) (1 + t) 1 + t

cos2 = cosA - sinA

Divide through by 2

:. cos = cos - sin sub in t for

1 t 1 - t= ---------- - ---------- :. cos = --------

(1 + t) (1 + t) 1 + t

2tan

tan2 = ------------1 - tan

Divide through by 2

2tan

:. tan = ----------------sub in t for1 - tan

2t:. tan = --------

1 - t

X-

2

X-

2

-

2

-

2

-

2

-

2

-

2

-

2

-

2

-

2

-

2


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6.8: Solving using Transformation (Subsidiary / Auxiliary Angles) (Extension 1)

b btan = --- similarly: sin = -----------

a (a + b)

acos = -----------

(a + b)

Let (a + b) = R

:. R . sin = b

R . cos = a

Show that: R . sin ( + ) = (a . sin ) + (b . cos) Proof

LHS: R . sin ( + )

= R . (sin . cos ) + R . (cos . sin)

= a . sin + b . cos = RHS

:. R . sin ( + ) = (a . sin) + (b . cos)

Where (a + b) = R, and is in the first quadrant

Furthermore:

~ R . sin ( - ) = (a . sin) - (b . cos)

~ R . cos ( + ) = (a . cos) - (b . sin)

~ R . cos ( - ) = (a . cos) + (b . sin)

e.g: solve 3 . sin + cos = 1 where 0 360

LHS: 3 . sin + cos note: R = (a + b) = (3 + 1)

= 2sin ( + 30) = 1 a = 3 = 2

sin ( + 30) = 0.5 * b = 1

:. = 0, 120, 360 b 1note: tan = --- = --- =

30a 3

* find where ( + 30) = 0.5i.e: quadrants 1 and 2 (and 360)

6.9: Products as Sums and Differences (Extension 1)

b

a

(a +b)


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Using the addition theorems:

1. sin (A + B) = (sinA . cosB) + (sinB . cosA)

2. sin (A B) = (sinA . cosB) (sinB . cosA)

3. cos (A + B) = (cosA . cosB) (sinA . sinB)

4. cos (A B) = (cosA . cosB) + (sinA . sinB)

~ Add 1 + 2

:. 2 (sinA . cosB) = sin (A + B) + sin (A B)

= sin (sum) + sin (difference)

~ Subtract 2 1

:. 2 (cosA . sinB) = sin (A + B) sin (A B)

= sin (sum) sin (difference)

~ Add 3 + 4:. 2 (cosA . cosB) = cos (A + B) + cos (A B)

= cos (sum) + cos (difference)

~ Subtract 4 3

:. 2 (sinA . sinB) = cos (A B) cos (A + B)

= cos (difference) cos (sum)

6.10: Sums or Differences as Products (Extension 2)

5.

6 5. .

7.

8.


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Using the product theorems:

5. 2 (sinA . cosB) = sin (A + B) + sin (A B)

6. 2 (cosA . sinB) = sin (A + B) sin (A B)

7. 2 (cosA . cosB) = cos (A + B) + cos (A B)

8. 2 (sinA . sinB) = cos (A B) cos (A + B)

~ let U = (A + B);V = (A B)

U + V U - V:. A = -------- B = --------

2 2

Substitute values of U and V into 5 8

U + V U - V9. sinU + sinV = 2sin -------- cos --------

2 2

U V U + V10. sinU sinV = 2sin -------- cos --------

2 2

U + V U - V11. cosU + cosV = 2cos -------- cos --------

2 2

U + V U - V12. a) cosV cosU = 2sin -------- sin --------

2 2

U + V U - Vb) cosU cosV = -2sin -------- sin --------

2 2

e.g: express sin6x sin4xin terms of products

6x 4x 6x + 4xsin6x sin4x = 2sin ( ---------- ) . cos ( ---------- )

2 2

2x 10x= 2sin ----- . cos ----- = 2sinx . cos5x

2 2


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Mathematics Extension 1 Notes

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