further trigonometric identities and their applications
TRANSCRIPT
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Further Trigonometric identities and their applications
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What trigonometric identities have we learnt so far?
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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
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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.
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7.1 Addition formulae
π .πππ ( π¨βπ© )β‘ππππ¨ππππ©+ππππ¨ππππ©Show that:
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7.1 Addition formulae
π .πππ ( π¨+π©)β‘ππππ¨ππππ©βππππ¨ππππ©
Show that:
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7.1 Addition formulae
π .πππ ( π¨+π© )β‘ππππ¨ππππ©+ππππ¨ππππ©
Show that:
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7.1 Addition formulae
4
Show that:
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7.1 Addition formulae
π .πππ ( π¨+π©)β‘ ππππ¨+ππππ©πβππππ¨ππππ©
Show that:
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7.1 Addition formulae
π .πππ ( π¨βπ©)β‘ ππππ¨β ππππ©π+ππππ¨ππππ©
Show that:
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7.1 Addition formulaeShow that:
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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)
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7.1 Addition formulae9. Given that 2 3
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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.
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7.2 Double angle formulae
π .ππππ π¨β‘π ππππ¨ππππ¨Show that:
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7.2 Double angle formulae
β 1
Show that:
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7.2 Double angle formulae
π .ππππ π¨β‘πππππ¨
πβ ππππ π¨
Show that:
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7.2 Double angle formulae
π .π ππππ½ππππ
π½π
Rewrite the following expressions as a single trigonometric function:
b
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7.2 Double angle formulae
π .πππππ
Given that , and that find the exactvalues of
b
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7.3 Using double angle formulae to solve more equations and prove more identities
1. Prove the identity
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7.3 Using double angle formulae to solve more equations and prove more identities
2. By expanding
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7.3 Using double angle formulae to solve more equations and prove more identities
3. Given that and express
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7.3 Using double angle formulae to solve more equations and prove more identities
4. Solve .
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Find the maximum value of .
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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.
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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
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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
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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.
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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 .
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7.5 Factor Formulae
1. Use the formulae for and to derive the result that .
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7.5 Factor Formulae
2. Using the result that . a. show that b. solve, for ,
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7.5 Factor Formulae
3. Prove that .