A new satellite network tower was installed in the non receptive areas of mabatang and calaguiman.the satellite in branched by two wires fastened to the ground at the same point.the shorter wire is 30 ft long and the fastened to the tower 10 feet above the foot of the tower.the longer one is 33 ft long and is fastened to the tower 17 ft above the foot of the pole. Find the exact value of sin x-y, the sine angle between two wires.and find CSC x and sec y?
Please answer my problem question
Did you make a sketch?
I have 2 right-angled triangles, with a common base of k ft
I see two sine values:
sin x = 17/33
sin y = 10/30
using the smaller triangle: k^2 + 10^2 = 30^2
k = √800 = 20√2
using the larger triangle: k^2 + 17^2 = 33^2
k = √800 = 20√2 , well that is good, they are the same.
so cosx = 20√2/33
cosy = 20√2/30
sin(x-y) = sinx cosy - cosx siny
= (17/33)(20√2/30) - (20√2/33)(10/30)
= 34√2/99 - 20√2/99
= (14√2)/99
cscx = 1/sinx = 1/(17/33) = 33/17
secy = 1/cosy = 30 / (20√2)
sinx = 17/33, so cosx = √800/30
siny = 10/30, so cosy = √800/33
Now just use your trig formulas.
sin(x-y) = sinx cosy - cosx siny
cscx = 1/sinx
secy = 1/cosy
To find the exact value of sin(x-y), we first need to find the values of x and y.
Let's start by drawing a diagram of the situation described:
A (Tower Top)
/|
/ | 33 ft (Longer Wire)
/ |
/ |
/ |
/ C
10 ft (Shorter Wire)
We can use the trigonometric ratios tangent (tan) and sine (sin) to find the angles.
First, let's find the angle x:
We can use the tangent (tan) ratio:
tan(x) = opposite/adjacent
tan(x) = 10/30
tan(x) = 1/3
To find the angle x, we can use the inverse tangent function (tan^(-1)) or arc tangent (atan) on a calculator.
x = atan(tan(x))
x = atan(1/3)
x ≈ 18.4349°
Next, let's find the angle y:
We can calculate the angle y using the same steps:
tan(y) = 10/33
tan(y) ≈ 0.3030
y = atan(tan(y))
y = atan(0.3030)
y ≈ 17.3791°
Now that we have the values of x and y, we can find sin(x-y):
sin(x-y) = sin(x) * cos(y) - cos(x) * sin(y)
We can find the values of sin(x) and cos(y) using the sin and cos functions on a calculator:
sin(x) ≈ sin(18.4349°) ≈ 0.3249
cos(y) ≈ cos(17.3791°) ≈ 0.9551
Similarly, we can find cos(x) and sin(y):
cos(x) ≈ cos(18.4349°) ≈ 0.9455
sin(y) ≈ sin(17.3791°) ≈ 0.3043
Plugging these values in, we get:
sin(x-y) ≈ (0.3249 * 0.9551) - (0.9455 * 0.3043)
sin(x-y) ≈ 0.3109
Therefore, the exact value of sin(x-y) is approximately 0.3109.
Now, let's find the values of CSC(x) and SEC(y):
CSC(x) is the reciprocal of SIN(x):
CSC(x) = 1 / sin(x)
CSC(x) ≈ 1 / 0.3249
CSC(x) ≈ 3.0761
SEC(y) is the reciprocal of COS(y):
SEC(y) = 1 / cos(y)
SEC(y) ≈ 1 / 0.9551
SEC(y) ≈ 1.0473
Therefore, CSC(x) is approximately 3.0761 and SEC(y) is approximately 1.0473.