1. use the suitable product-to-sum formula to rewrite the following expression: sin(13x/2)sin(5x/2)
2. Find value of cos(255degrees)cos(105degrees)
is it 1/4?
3. cos(pi/12) - cos(5pi/12)
root3/4?
4. Use the appropriate sum-to-product formula to rewrite the expression sin6x - sin9x
I don't really understand how to do these, but I got -2sin(3x/2)cos(15x/2)..
5. same type of question: rewrite the expression cos4x - cos3x
cosx??
Look at Theorem 1.6 about 3/4 of the way down this British webpage for some lesser know identities
http://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec2.pdf
so comparing your
sin(13x/2)sin(5x/2)
with
cosA - cosB = -sin((A+B)/2)sin((A-B)/2))
cosB - cosA = sin((x+y)/2)sin((x-y)/2))
and setting
(A+B)/2= 13x/2---> A+B = 13x
(A-B)/2 = 5x/2----> A-B = 5x
adding them
2A = 18x
A = 9x
B = 4x
sin(13x/2)sin(5x/2) = cos(4x) - cos(9x)
2.
Use the appropriate formula from the same section
or
cos(255) = -cos(75)
= -cos(45+30)
= -(cos45cos30 - sin45sin30)
= -( (√2/2)(√3/2) - (√2/2)(1/2)) = -(√6/4 - √2/4)
= √2/4 - √6/4
cos(105) = -cos75 = √2/4 - √6/4
cos(255degrees)cos(105degrees) = (√2-√6)^2/16
= (2 - 2√12 + 6)/16
= (8 - 4√3)/16 = (2 - √3)/4
try the others
1. To rewrite the expression sin(13x/2)sin(5x/2) using the suitable product-to-sum formula, we can use the formula sin(a)sin(b) = (1/2)(cos(a - b) - cos(a + b)).
In this case, a = 13x/2 and b = 5x/2.
So, substituting the values into the formula, we get:
sin(13x/2)sin(5x/2) = (1/2)(cos(13x/2 - 5x/2) - cos(13x/2 + 5x/2))
Simplifying further:
= (1/2)(cos(8x/2) - cos(18x/2))
= (1/2)(cos(4x) - cos(9x))
2. To find the value of cos(255 degrees)cos(105 degrees), first, convert the angles to radians since trigonometric functions usually work in radians.
255 degrees converted to radians = 255 * (π/180) = 17π/12
105 degrees converted to radians = 105 * (π/180) = 7π/12
Now, using the product-to-sum formula, cos(a)cos(b) = (1/2)(cos(a + b) + cos(a - b)), with a = 17π/12 and b = 7π/12:
cos(17π/12)cos(7π/12) = (1/2)(cos(17π/12 + 7π/12) + cos(17π/12 - 7π/12))
Simplifying further:
= (1/2)(cos(24π/12) + cos(10π/12))
= (1/2)(cos(2π) + cos(5π/6))
= (1/2)(1 + (√3)/2)
= 1/4 + (√3)/4
So, cos(255 degrees)cos(105 degrees) is equal to 1/4 + (√3)/4.
3. To evaluate cos(pi/12) - cos(5pi/12), first, convert the angles to radians.
pi/12 and 5pi/12 are already in radians.
Using the sum-to-product formula, cos(a) - cos(b) = -2sin((a + b)/2)sin((a - b)/2):
cos(pi/12) - cos(5pi/12) = -2sin((pi/12 + 5pi/12)/2)sin((pi/12 - 5pi/12)/2)
Simplifying further:
= -2sin(6pi/24)sin(-2pi/24)
= -2sin(pi/4)sin(-pi/12)
= -2(1/√2)(-1/2)
= √3/2
So, cos(pi/12) - cos(5pi/12) is equal to √3/2.
4. To rewrite the expression sin(6x) - sin(9x) using the appropriate sum-to-product formula, we can use the formula sin(a) - sin(b) = 2cos((a + b)/2)sin((a - b)/2).
In this case, a = 6x and b = 9x.
So, substituting the values into the formula, we get:
sin(6x) - sin(9x) = 2cos((6x + 9x)/2)sin((6x - 9x)/2)
Simplifying further:
= 2cos(15x/2)sin(-3x/2)
= -2cos(15x/2)sin(3x/2)
So, sin(6x) - sin(9x) can be rewritten as -2cos(15x/2)sin(3x/2).
5. To rewrite the expression cos(4x) - cos(3x) using the appropriate sum-to-product formula, we can use the formula cos(a) - cos(b) = -2sin((a + b)/2)sin((a - b)/2).
In this case, a = 4x and b = 3x.
So, substituting the values into the formula, we get:
cos(4x) - cos(3x) = -2sin((4x + 3x)/2)sin((4x - 3x)/2)
Simplifying further:
= -2sin(7x/2)sin(x/2)
So, cos(4x) - cos(3x) can be rewritten as -2sin(7x/2)sin(x/2).