Further Trigonometric identities and their applications

Further Trigonometric identities and their applications

What trigonometric identities have we learnt so far?

Trigonometric identities learnt so far

𝟏 .𝒕𝒂𝒏𝜽=𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽

(𝒂𝒔𝒚𝒎𝒑𝒐𝒕𝒐𝒕𝒆𝒔 :𝜽=𝟗𝟎°+𝟏𝟖𝟎°𝒏)

5

𝟐 .𝒄𝒐𝒕 𝜽=𝒄𝒐𝒔𝜽𝒔𝒊𝒏𝜽

(𝒂𝒔𝒚𝒎𝒑𝒐𝒕𝒐𝒕𝒆𝒔 :𝜽=𝟏𝟖𝟎°𝒏)

𝟑 .𝒔𝒆𝒄 𝜽=𝟏

𝒄𝒐𝒔𝜽(𝒂𝒔𝒚𝒎𝒑𝒕𝒐𝒕𝒆𝒔 :𝜽=𝟗𝟎°+𝟏𝟖𝟎°𝒏)

𝟒 .𝒄𝒐𝒔𝒆𝒄 𝜽=𝟏

𝒔𝒊𝒏𝜽(𝒂𝒔𝒚𝒎𝒑𝒕𝒐𝒕𝒆𝒔 :𝜽=𝟏𝟖𝟎°𝒏)

6

7

= 90-)

= 90-)

= - sin

=

= - tan

7.1 Addition formulae

𝟏 .𝒔𝒊𝒏 ( 𝑨+𝑩 )≡𝒔𝒊𝒏𝑨𝒄𝒐𝒔𝑩+𝒄𝒐𝒔𝑨𝒔𝒊𝒏𝑩

𝟐 .𝒔𝒊𝒏 ( 𝑨−𝑩)≡𝒔𝒊𝒏𝑨𝒄𝒐𝒔𝑩−𝒄𝒐𝒔𝑨 𝒔𝒊𝒏𝑩

𝟑 .𝒄𝒐𝒔 ( 𝑨+𝑩)≡𝒄𝒐𝒔𝑨𝒄𝒐𝒔𝑩−𝒔𝒊𝒏𝑨𝒔𝒊𝒏𝑩

𝟒 .𝒄𝒐𝒔 ( 𝑨−𝑩)≡𝒄𝒐𝒔𝑨𝒄𝒐𝒔𝑩+𝒔𝒊𝒏𝑨𝒔𝒊𝒏𝑩

𝟓 .𝒕𝒂𝒏 ( 𝑨+𝑩)≡ 𝒕𝒂𝒏𝑨+𝒕𝒂𝒏𝑩𝟏−𝒕𝒂𝒏𝑨𝒕𝒂𝒏𝑩

𝟔 .𝒕𝒂𝒏 ( 𝑨−𝑩)≡ 𝒕𝒂𝒏𝑨− 𝒕𝒂𝒏𝑩𝟏+𝒕𝒂𝒏𝑨𝒕𝒂𝒏𝑩

You need to know and be able to use the addition formulae.

7.1 Addition formulae

𝟏 .𝒄𝒐𝒔 ( 𝑨−𝑩 )≡𝒄𝒐𝒔𝑨𝒄𝒐𝒔𝑩+𝒔𝒊𝒏𝑨𝒔𝒊𝒏𝑩Show that:

7.1 Addition formulae

𝟐 .𝒄𝒐𝒔 ( 𝑨+𝑩)≡𝒄𝒐𝒔𝑨𝒄𝒐𝒔𝑩−𝒔𝒊𝒏𝑨𝒔𝒊𝒏𝑩

Show that:

7.1 Addition formulae

𝟑 .𝒔𝒊𝒏 ( 𝑨+𝑩 )≡𝒔𝒊𝒏𝑨𝒄𝒐𝒔𝑩+𝒄𝒐𝒔𝑨𝒔𝒊𝒏𝑩

Show that:

7.1 Addition formulae

4

Show that:

7.1 Addition formulae

𝟓 .𝒕𝒂𝒏 ( 𝑨+𝑩)≡ 𝒕𝒂𝒏𝑨+𝒕𝒂𝒏𝑩𝟏−𝒕𝒂𝒏𝑨𝒕𝒂𝒏𝑩

Show that:

7.1 Addition formulae

𝟔 .𝒕𝒂𝒏 ( 𝑨−𝑩)≡ 𝒕𝒂𝒏𝑨− 𝒕𝒂𝒏𝑩𝟏+𝒕𝒂𝒏𝑨𝒕𝒂𝒏𝑩

Show that:

7.1 Addition formulaeShow that:

7.1 Addition formulae8. Given that and 180 and B is obtuse, find the value of

a. cos (A – B)b. tan (A + B)

7.1 Addition formulae9. Given that 2 3

7.2 Double angle formulae

𝟏 .𝒔𝒊𝒏𝟐 𝑨≡𝟐 𝒔𝒊𝒏𝑨𝒄𝒐𝒔𝑨 – 1

𝟑 .𝒕𝒂𝒏𝟐 𝑨≡𝟐𝒕𝒂𝒏𝑨

𝟏− 𝒕𝒂𝒏𝟐 𝑨

You need to know and be able to use the double angle formulae.

7.2 Double angle formulae

𝟏 .𝒔𝒊𝒏𝟐 𝑨≡𝟐 𝒔𝒊𝒏𝑨𝒄𝒐𝒔𝑨Show that:

7.2 Double angle formulae

– 1

Show that:

7.2 Double angle formulae

𝟑 .𝒕𝒂𝒏𝟐 𝑨≡𝟐𝒕𝒂𝒏𝑨

𝟏− 𝒕𝒂𝒏𝟐 𝑨

Show that:

7.2 Double angle formulae

𝒂 .𝟐 𝒔𝒊𝒏𝜽𝟐𝒄𝒐𝒔

𝜽𝟐

Rewrite the following expressions as a single trigonometric function:

b

7.2 Double angle formulae

𝒂 .𝒔𝒊𝒏𝟐𝒙

Given that , and that find the exactvalues of

b

7.3 Using double angle formulae to solve more equations and prove more identities

1. Prove the identity

7.3 Using double angle formulae to solve more equations and prove more identities

2. By expanding

7.3 Using double angle formulae to solve more equations and prove more identities

3. Given that and express

7.3 Using double angle formulae to solve more equations and prove more identities

4. Solve .

Find the maximum value of .

7.4 Write as a sine function or cosine function only

1. Show that you can express in the form R, where , , giving your values of and to 1 decimal place where appropriate.

7.4 Write as a sine function or cosine function only

2. a. Show that you can express in the form R, where , . b. Hence sketch the graph of

7.4 Write as a sine function or cosine function only

3. a. Express in the form R, where , O. b. Hence sketch the graph of

7.4 Write as a sine function or cosine function only

4. Without using calculus, find the maximum value of , and give the smallest positive value of at which it arises.

7.4 Write as a sine function or cosine function only

For positive values of a and b,

can be expressed in the form with R>0 and

can be expressed in the form (θ) with R>0 and

where = a and = b

and .

7.5 Factor Formulae

1. Use the formulae for and to derive the result that .

7.5 Factor Formulae

2. Using the result that . a. show that b. solve, for ,

7.5 Factor Formulae

3. Prove that .