Further Trigonometric identities and their applications
What trigonometric identities have we learnt so far?
Trigonometric identities learnt so far
π .ππππ½=ππππ½ππππ½
(πππππππππππ :π½=ππΒ°+πππΒ°π)
5
π .πππ π½=ππππ½ππππ½
(πππππππππππ :π½=πππΒ°π)
π .πππ π½=π
ππππ½(ππππππππππ :π½=ππΒ°+πππΒ°π)
π .πππππ π½=π
ππππ½(ππππππππππ :π½=πππΒ°π)
6
7
= 90-)
= 90-)
= - sin
=
= - tan
7.1 Addition formulae
π .πππ ( π¨+π© )β‘ππππ¨ππππ©+ππππ¨ππππ©
π .πππ ( π¨βπ©)β‘ππππ¨ππππ©βππππ¨ ππππ©
π .πππ ( π¨+π©)β‘ππππ¨ππππ©βππππ¨ππππ©
π .πππ ( π¨βπ©)β‘ππππ¨ππππ©+ππππ¨ππππ©
π .πππ ( π¨+π©)β‘ ππππ¨+ππππ©πβππππ¨ππππ©
π .πππ ( π¨βπ©)β‘ ππππ¨β ππππ©π+ππππ¨ππππ©
You need to know and be able to use the addition formulae.
7.1 Addition formulae
π .πππ ( π¨βπ© )β‘ππππ¨ππππ©+ππππ¨ππππ©Show that:
7.1 Addition formulae
π .πππ ( π¨+π©)β‘ππππ¨ππππ©βππππ¨ππππ©
Show that:
7.1 Addition formulae
π .πππ ( π¨+π© )β‘ππππ¨ππππ©+ππππ¨ππππ©
Show that:
7.1 Addition formulae
4
Show that:
7.1 Addition formulae
π .πππ ( π¨+π©)β‘ ππππ¨+ππππ©πβππππ¨ππππ©
Show that:
7.1 Addition formulae
π .πππ ( π¨βπ©)β‘ ππππ¨β ππππ©π+ππππ¨ππππ©
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 β B)b. tan (A + B)
7.1 Addition formulae9. Given that 2 3
7.2 Double angle formulae
π .ππππ π¨β‘π ππππ¨ππππ¨ β 1
π .ππππ π¨β‘πππππ¨
πβ ππππ π¨
You need to know and be able to use the double angle formulae.
7.2 Double angle formulae
π .ππππ π¨β‘π ππππ¨ππππ¨Show that:
7.2 Double angle formulae
β 1
Show that:
7.2 Double angle formulae
π .ππππ π¨β‘πππππ¨
πβ ππππ π¨
Show that:
7.2 Double angle formulae
π .π ππππ½ππππ
π½π
Rewrite the following expressions as a single trigonometric function:
b
7.2 Double angle formulae
π .πππππ
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 (ΞΈ) 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 .
Recommended
