Could someone please post a solution to these 4 questions:
Hello,
to #3.a)
(For some reason the Latex doesn't work properly anymore. I'll repeat the equations in plain text)
cos(2x) = 0 or cos(2x) = -1 or cos(2x) = 1
This character π means pi
From the 1rst equ.:
2x = -π/2 --> x = -π/4 or
2x = -3π/2 ---> x = -3π/4
from the 2nd equ.:
2x = -π ---> x = -π/2
from the 3rd equ.:
2x = 0 ---> x = 0 or
2x = -2π ---> x = -π
EB
Hello, SportfreundeKeaneKent!
Here's the very first one . . .
1. If cos(x) = -12/13, where π < x < 3π/2
then find the exact value of sin(2x)
We'll use the identity: .sin(2x) .= .2·sin(x)·cos(x)
We know that: cos(x) = -12/13 . . . we need sin(x).
Angle x is in Quadrant 3.
Since cos(x) = -12/13 = adj/hyp,
. . we use Pythagorus and find that: opp = -5
Hence: .sin(x) = -5/13
Therefore: .sin(2x) .= .2(-5/13)(-12/13) .= .120/169
Hello again, SportfreundeKeaneKent!
Here's #4(b) . . .
We will use these identities:Prove:
. . . . . . . . . . . . . .tan(x) - tan(y)
. . tan(x - y) . = . --------------------
. . . . . . . . . . . . .1 + tan(x)·tan(y)
. . sin(x - y) .= .sin(x)·cos(y) - sin(y)·cos(x)
. . cos(x - y) .= .cos(x)·cos(y) + sin(x)·sin(y)
. . . . . . . . . . . sin(x - y) . . . . . sin(x)·cos(y) - sin(y)·cos(x)
tan(x - y) . = . ------------ . = . -----------------------------------
. . . . . . . . . . . cos(x - y) . . . . .cos(x)·cos(y) + sin(x)·sin(y)
Divide top and bottom by cos(x)·cos(y):
. . .sin(x)·cos(y) . .sin(y)·cos(x) . . . . . . sin(x) . sin(y)
. . .--------------- - --------------- . . . . . . ------- - -------
. . .cos(x)·cos(y) . cos(x)·cos(y) . . . . . . cos(x) . cos(y)
. . ------------------------------------ . = . ---------------------
. . .cos(x)·cos(y) . .sin(x)·sin(y) . . . . . . . . sin(x) sin(y)
. . .--------------- + --------------- . . . . .1 + -------·--------
. . .cos(x)·cos(y) . .cos(x)·cos(y) . . . . . . . .cos(x) cos(y)
. . . . . . . tan(x) - tan(y)
. . . = . ---------------------
. . . . . . 1 + tan(x)·tan(y)
Hello, Sportsfreund,
#2
Use the property:
As Soroban has demonstrated use the Pythagoran theorem:
. Because x is in the 4. quadrant the sine-value must be negative.
. Because w is in the 2. quadrant the cosine-value must be negative.
Now you know all values necessary:
EB
Hello, SportfreundeKeaneKent!
There's something wrong with 4(a).
. . Is there a typo?
The right side is always equal to 1.
. . So it's a silly way to write a problem.
Besides, the statement is not an identity.
Try x = π/3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._
The left side is: .sec(π/3)[1 - sin(π/3)] .= .2(1 - √3/2) . ≠ . 1