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The sine of the Sum of 2 Angles:. sin (A + B) = sin A cos B + cos A sin B. Aim: How do we find the trig values of double angles?. Do Now:. Use. to prove sin 2A = 2 sin A cos A. Sine of Double Angles. sin 2A = 2 sin A cos A. sin (A + B) = sin A cos B + cos A sin B. sin 2A = sin(A + A). - PowerPoint PPT Presentation
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Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Aim: How do we find the trig values of double angles?
Do Now:Use
The sine of the Sum of 2 Angles:sin (A + B) = sin A cos B + cos A sin B
to provesin 2A = 2 sin A cos A
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Sine of Double Angles
sin (A + B) = sin A cos B + cos A sin B
sin 2A = 2 sin A cos A
sin 2A = sin(A + A)= sin A cos A + cos A sin A
sin 2A = 2sin A cos A
ex. show that sin 90 = 1 by using sin 2(45).sin 2(45).
sin 2(45) = 2 sin 45 cos 45
2 2
2
2
2
4
41
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Cosine of Double Angles
Prove cos 2A = cos2A – sin2AThe Cosine of the Sum of 2 Angles:
cos (A + B) = cos A cos B – sin A sin B
cos 2A = cos A cos A – sin A sin A
cos 2A = cos2 A – sin2 A
cos 2A = 2cos2A – 1
also
and cos 2A = 1 – 2sin2 A
Prove these two
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Tangent of Double Angles
tan(A B) tanA tanB
1 tanAtanB
tan2A2tanA
1 tan2 AProve
tan(A A) tanA tanA
1 tanAtanA
tan2A2tanA
1 tan2 A
ex. Show tan 120 by using tan 2(60)
3
tan2(60) 2tan60
1 tan2 60
2 3
1 ( 3)2
2 3
1 ( 3)2
2 3
1 3
2 3
2
3
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
The Trigonometry of Double Angles
tan2A2tanA
1 tan2 A
sin 2A = 2sin A cos A
cos 2A = cos2 A – sin2 A
cos 2A = 2cos2A – 1 cos 2A = 1 – 2sin2 A
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Model Problems
Cos A = -5/13, and A is the measure of an angle in QII.
a. Find sin 2Ab. Find cos 2Ac. Find tan 2Ad. Determine the quadrant in which
the angle whose measure is 2A lies.
a) Find sin AQII - sin A is positive
sin2 A = 1 – cos2 Asin2 A = 1 – (-5/13)2
sin2 A = 1 – 25/169sin2 A = 144/169sin A = 12/13
sin 2A = 2sin A cos A
sin 2A = 2(12/13)(-5/13) = -120/169
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Model Problems
Cos A = -5/13, and A is the measure of an angle in QII.
b. Find cos 2A
b) cos 2A = cos2 A – sin2 A
cos 2A = (-5/13)2 – (12/13)2
cos 2A = 25/169 – 144/169
cos 2A = - 119/169
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Model Problems
Cos A = -5/13, and A is the measure of an angle in QII.
c. Find tan 2A
c) Find tan A
tanA sinA
cosA
tanA
1213 513
12
5
tan2A2tanA
1 tan2 A
tan2A2
12
5
112
5
2
tan2A120
119
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Model Problems
Cos A = -5/13, and A is the measure of an angle in QII.
d. Determine the quadrant in whichthe angle whose measure is 2A lies.
cos 2A = - 119/169
sin 2A = -120/169
bothnegative
only occurs in QIII
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Regents Question
2cos
sin2
The expression is equivalent to
(3) cot
(2) sec (4) sin
(1) csc
2cos
sin2
2cos
2sin cos
sin 2A = 2sin A cos A
1csc
sin
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Regents Question
2If sin A = where 0 < A < 90, what is the
3value of sin 2A?
2 5 4 51) 3)
3 9
2 5 4 52) 4) -
9 9
sin 2A = 2sin A cos A
sin 2A =
2 5sin cos
3 3A A
2 52
3 3
4 5
9
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
Regents Question2
2 2
2 2
The expression cos cos2 is equivalent to
1) sin 3) cos 1
2) -sin 4) -cos 1
cos 2A = cos2 A – sin2 A
2cos cos2 2 2 2cos cos sin
2sin
2 2 2cos cos sin
Aim: Trig of Double Angles Course: Alg. 2 & Trig.
The Product Rule
