<u>Answer</u>
-3/√13
<u>Explanation</u>
We are going to take angle ∅ to be at the origin (point (0,0)).
The point at (-2, -3) means, 2 units to the left (negative side of x-axis) and 3 units downwards (negative side of y-xis).
This will make a right triangle with a hypotenuse of √(2²+3²) = √13
The angle ∅ is in the 3rd quadrant where cosine is negative.
cosine = adjacent/hypotenuse
The exact answer for cos ∅ = -2/√13
Answer:
6*5*4*3*2*1+4*3*2*1-3*2*1 =
6*5*4*3*2*1+(4-1)*3*2*1=
6*5*4*3*2*1+3*3*2*1=
3*2*1*(6*5*4+3)
this number has more than 2 factors, therefore is composite
If the terminal ray of an angle passes through (-5, -3), that means it passes through x coordinate -5 and y coordinate -3. Both x and y are negative in only one quadrant and that is the third one. If we plot the point (-5, -3) in the coordinate plane in Q3, and then draw a vertical line to connect that point to the negative x axis and then draw a line to connect that point to the origin, what we have created is right triangle with a base of -5 and a height of -3. The tangent ratio relates the side opposite the reference angle to the side adjacent to the reference angle. We have both of those measures already and do not have a need for the length of the hypotenuse since the tangent ratio doesn't use it. The tangent of the reference angle is -3/-5 which is 3/5.
The local minimum of function is an argument x for which the first derivative of function g(x) is equal to zero, so:
g'(x)=0
g'(x)=(x^4-5x^2+4)'=4x^3-10x=0
x(4x^2-10)=0
x=0 or 4x^2-10=0
4x^2-10=0 /4
x^2-10/4=0
x^2-5/2=0
[x-sqrt(5/2)][x+sqrt(5/2)]=0
Now we have to check wchich argument gives the minimum value from x=0, x=sqrt(5/2) and x=-sqrt(5/2).
g(0)=4
g(sqrt(5/2))=25/4-5*5/2+4=4-25/4=-9/4
g(-sqrt(5/2))=-9/4
The answer is sqrt(5/2) and -sqrt(5/2).