Few trig problems?

Question

1. "Simplify: ((sin x)/(1 +cos x)) + ((sin x)/(1-cos x))"
2. Find the sum of the reciprocals of two numbers whose sum is 8 and whose product is 16

Answer 1

1) answer = 2 (get like denominators and simplify)
2) use the equations:

(1/x) + (1/y) = 8
(1/x)(1/y) = 16 (use this one to solve for y in terms of x, and plug that value into first equation to solve for x and then go back and get y)

answer = don't feel like solving

Answer 2

1. ((sin x)/(1 +cos x)) + ((sin x)/(1-cos x))
((sin x)(1 - cos x)/(1 +cos x)(1 - cos x)) + ((sin x)(1 + cos x)/(1-cos x)(1 + cos x))"
(sin x - sin x cos x + sin x + sin x cos x)/((1 + cos x)(1 - cos x))
2sin x/(1 - cos^2 x)
2sin x/sin^2 x

2/sin x

2. Two numbers whose sum is 8 and whose product is 16.

x + y = 8
xy = 16

y = 8 - x
x(8 - x) = 16
8x - x^2 = 16
0 = x^2 - 8x + 16
0 = (x -4)^2
0 = x - 4
x = 4, y = 4

1/4 + 1/4 = 2/4 = 1/2

Answer 3

#2 is not trig.

One number is x, the other is y.

We have:
x+y=8
xy=16

So x=8-y

(8-y)y=16

y^2-8y+16=0

(y-4)^2=0

y=4

So x=y=4

The reciprocals are 1/4 and 1/4.

1/4+1/4=1/2

#1
We make a common denominator:

[sin x (1-cos x)+sin x (1+cos x)]/(1+cos(x))(1-cos(x))

=2sin(x)/(1-cos^2(x))

=2sin(x)/sin^2(x)

=2/sin(x)=2csc(x)

Answer 4

1. Factor sin(x)
sin(x)*(1/(1+cos(x))+1/(1-cos(x))) = sin(x)*(1-cos(x)+1+cos(x))/((1-cos(x))*(1+cos(x))) = sin(x)*(2)/(1-cos^2(x)) = 2sin(x)/sin^2(x) = 2/sin(x)

2. x+y =8
xy =16

y =16/x ---> x + 16/x = 8 ---> x^2 -8x +16 = 0

Factor (x-4)*(x-4) = 0 therefore x = 4 and y = 4

So 4+1/4 = 17/4

Answer 5

1. Multiply top and bottom of first by (1-cos(x)) and second by (1+cos(x)):
((sin(x))(1-cos(x)))/(1-cos^2(x)) + ((sin(x))(1+cos(x)))/(1-cos^2(x))
Use trig id to know that 1-cos^2(x) = sin^2(x) and cancel the sinx:
(1-cos(x))/sin(x) + (1+cos(x))/sin(x)
Simplify:
Answer = 2/sin(x) = 2csc(x)

2. 4+4=8, 4*4=16, 1/4 + 1/4 = 2/4
Answer = 1/2

Answer 6

x = rcos? y = rsin? So in simple terms pass get your calculator and plug interior the numbers of your issues into (r cos?, r sin?) and you gets your x,y coordinates. make certain your calculator is in DEG mode. If that's in RAD mode you will get incorrect solutions.