5-4
Multiple-Angle Identities
Trig Identities Song
To the tune of Rudolph the Red-Nosed Reindeer
You know reciprocal and quotient and cofunction and odd/even
We’ve also done the sum or the difference and pythagorean
But do you recall the most challenging identities of all?
First we have double angle, don’t forget that cosine has 3,
Next are the power-reducing, you need time to memorize these.
Lastly are the half angle, these all involve a square root –
The sheet Mrs. Mullen just gave you should help you do these proofs.
It’s almost time for Christmas Eve, things will get easier by then –
We’ll do Laws of Sines and Cosines, finish with limits. The end!
I challenge you to keep trying… these lovely trig identities
They’ll help you be problem solvers and maybe YOU’LL go down in history!
Recall sums & differences
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cos (q  a) = cos q cos a  sin q sin a
sin (q  a) = sin q cos a  cos q sin a
tan (q  a) = tan q  tan a
1  tan q tan a
Group Work – Sine of a
Double Angle
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Using the sine of a sum formula, substitute
in q, for both angles
Simplify the results
What is a formula for sin (q + q) of sin 2q?
Sine of a Double Angle Identity
Sin 2q = 2sinqcosq
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Ex 1 Write the expression as one
involving only sin q and cos q:
sin 2q + cos q
Group Work – Cosine of a
Double Angle
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Using the cosine of a sum formula,
substitute in q, for both angles
Simplify the results
What is a formula for cos (q + q) of cos 2q?
Using the Pythagorean theorem identities
for cos2q and sin2q, find two other formulas
for cos 2q
Cosine of a Double Identity
cos 2q = cos2q - sin2q
= 2cos2q - 1
= 1 – 2sin2q
 Ex 2 Find all the solutions to the
equation: cos 2x = cos x
Group Work – Tangent of a
Double Angle
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Using the tangent of a sum formula,
substitute in q, for both angles
Simplify the results
What is a formula for tan (q + q) of tan 2q?
Tangent of a Double Angle Identity
tan 2q =
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2tan q
1 – tan2 q
Ex 3 Prove 2csc 2u = csc2u tan u
Ex 4 Prove the identity
a)
cos 6x = 2cos2(3x) – 1
b)
sin 4x = 2 sin2x cos2x
Ex 5 Solve algebraically for exact solutions
in the interval [0,2p). Use your calculator
to support your work.
a)
cos 2x + cos x = 0
b)
sin 2x – cos 3x = 0
c)
cos 2x = sin x
Exit Ticket
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Write the expression as one involving
only sin q and cos q: sin 3q + cos 2q
Tonight’s Assignment
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P. 475 Ex 6 - 30 m. of 3
Study for Unit # 8 Test on Sums,
Differences and Multiple Angles