C3 Chapter 7
FURTHER TRIGONOMETRIC IDENTITIES
ADDITION FORMULAE
Remember that:
It is easily verified that sin(A + B) ≠ sinA + sin B
One counter-example is all that is needed
e.g.Let A = 30º and B = 45º
Then sin(A + B) = sin 75º = 0.9659…
and sin 30º + sin 45º = 1.2071…
so sin(30º + 45º) ≠ sin30º + sin45º
In fact:
(You do not need to know how to derive these formulae)
You can deduce the results from each other
Example 1
In sin(A + B) = sin A cos B + cos A sin B,
replace A with (90º - A)
Replace B by –B
Hence we have:
Example 2
Prove the identity
Example 3
Ex 7A p 99
DOUBLE ANGLE FORMULAE
In the formula sin (A + B) = sin A cos B + cos A sin B, let A = B
Therefore, sin (A + A) = sin A cos A + cos A sin A
sin 2A = 2 sin A cos A
Hence the sine double angle formula:
In the formula cos (A + B) = cos A cos B – sin A sin B, let A = B
Therefore, cos (A + A) = cos A cos A – sin A sin A
cos 2A = cos2A – sin2A
Since cos2A = 1 – sin2A, cos 2A = (1 – sin2A) - sin2A = 1 - 2sin2A
Or sin2A = 1 – cos2A, cos 2A = cos2A – (1 – cos2A) = 2cos2A – 1
Hence the 3 versions of the cosine double angle formula:
In the formula , let A = B
Hence the tangent double angle formula:
Example 1
Write the following as a single trigonometric ratio:
Example 2
Without using a calculator, find the value of
Example 3
Given that sin θ = ¼ and that θ is obtuse, find the exact value of sin 2θ
Ex 7B p 103
USING THE DOUBLE ANGLE FORMULAE TO PROVE IDENTITIES
Example
Prove the identitycosec θ – 2 cot 2θ cos θ = 2 sin θ
Ex 7C Q 1, 2, 7-13 p 106
TRIPLE ANGLE FORMULAE
HALF-ANGLE FORMULAE
These are useful in integration.
USING THE DOUBLE ANGLE FORMULAE TO SOLVE EQUATIONS
Example 1
Solve the equation3 sin 2θ = 4 tan θ for 0º ≤ θ ≤ 360º
Ex 7C Q 3, p 107
Example 2
Eliminate θ from the equations:
Note: θ is known as a parameter
Ex 7C Q 4-6, p 107
THE FORM “a cos θ ± b sin θ”
If you draw the graph of a cos θ + b sin θ it will have the same form as a sine or cosine graph.
It can be expressed in the form R sin (θ ± α) or R cos (θ ± α)
Example 1
Remember:
Ex 7D Q 1-7 p 111
This method can be used to find maximum or minimum values of expressions and to solve certain types of trig equations.
Example 2
Find the maximum value of 2 cos 2θ – 5 sin 2θ and the smallest positive value of θ for which it occurs.
Note; if necessary use a sketch to determine the max/min values
e.g.
Example 3
Solve the equation4 cos x + 3 sin x = 1 for 0º ≤ x ≤ 360º
Ex 7D Q 8-15 p 111
FACTOR FORMULAE
Replacing Q with –Q gives:
Hence:
sin P + sin Q = 2 x sin (semi-sum) cos (semi-difference)
sin P – sin Q = 2 x sin (semi-difference) cos (semi-sum)
Replacing Q with –Q gives exactly the same result, so this time we subtract the two starting equations:
Hence:
cos P + cos Q = 2 x cos (semi-sum) cos (semi-difference)
cos P – cos Q = - 2 x sin (semi-sum) sin (semi-difference)
Hence the 4 factor formulae:
Note the minus sign in the last formula
Example 1
Express as a sum or difference of sines:sin 4x cos 6x
Use
Example 2
Solve the equationcos 2θ + cos 3θ = 0 for 0 ≤ θ ≤ 2π
Ex 7E p 115
Mixed Ex 7F p 116
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JMcC