Precalculus section 6.3 Sum and Difference formulas Thm (Sum and

Precalculus
section 6.3
Sum and Difference formulas
Thm (Sum and Difference formulas)
(1)
cos (A  B) = cosAcosB  sinAsinB
(2)
cos (A  B) = cosAcosB  sinAsinB
(3)
sin (A  B) = sinAcosB  cosAsinB
(4)
sin (A  B) = sinAcosB  cosAsinB
A + tan B
(5)
tan (A  B) = tan
1tanAtanB
(6)
A  tan B
tan (A  B) = tan
1+ tanAtanB
Ex1,
Evaluate cos (15‰ )
Ex2,
Evaluate sin (15‰ )
Ex3,
Evaluate cos (105‰ )
Ex4,
Evaluate sin(15‰ )cos(30‰ ) + cos(15‰ )sin(30‰ )
(Ex5-Ex7, after inverse functions in section 6.6)
Ex5, Evaluate cos (sin1 ( 23 ) + cos1 ( 14 ) )
Ex6,
Evaluate sin (tan1 (2) + sec1 (3) )
Ex7,
Evaluate tan (tan1 ( 12 ) + cos1 ( 13 ) )
Ex8,
Prove cos( 321  )) =  sin )
Ex9,
Prove sin (x + 16 )  sin (x  16 ) = cos x
A cos x + B sin x
Thm: If A2 +B2 = 1, then A = cos α, and B = sin α for a unique α − [0, 21]
If A2 +B2 Á 1, then such α does not exist.
Form A cos x + B sin x
(a) A cos x + B sin x = A2 +B2 (  A2 2 cos x +  B2 2 sin x)
Let
A
A2 +B2
Let
A
A2 +B2
A +B
A +B
= cos α and  B2 2 = sin α
Note that cos2 α + sin2 α = 1
A +B
= A2 +B2 ( cos α cos x + sin α sin x) = A2 +B2 cos (x  α)
(b) A cos x + B sin x = A2 +B2 (  A2 2 cos x +  B2 2 sin x)
Ex10, f(x) = 12 cos x +
Ex11, f(x) =
3
2 cos
=
3
2
x 
A +B
B
sin " and  2 2 = cos "
A +B

2
2
= A +B ( sin " cos x +
sin x
1
2
sin x
Ex12, f(x) = cos x + 3 sin x
Ex13, g(x) = cos x + sin x
Ex14, f(x) = 3 cos x  sin x
A +B
Note that cos2 " + sin2 " = 1
cos " sin x) = A2 +B2 sin (x + " )