find each measure measurement indicated. Round your answers to the nearest tenth. Please show work​

9514 1404 393

Answer:

Step-by-step explanation:

1. The desired angle is given on the diagram as 125°.

__

For the rest of these problems, the Law of Sines applies. A side can be found from ...

  a = b(sin(A)/sin(B))

and an angle can be found from ...

  A = arcsin(a/b·sin(B))

__

2. Y = arcsin(y/z·sin(Z)) = arcsin(5/11·sin(88°))

  ∠Y = 27°

__

3. DF = (11 km)·sin(32°)/sin(103°)

  DF = 5.98 km ≈ 6 km

__

4. b = c·sin(B)/sin(C) = (13 ft)·sin(65°)/sin(180° -65° -37°)

  b = 12 ft

Answer:

#1: The measure of m<B is 125°.

#2: M<Y is equal a 27°.

#3: So the length of DF is 5.99 km.

#4: the side length AC is 13.1 feet.

Explanation:

# 1: The following given,

c = AB = 17 cm

a = BC = unknown

b = CA = 44 cm

Ø = 125

M<B means it is the angle at vertex B of the triangle, it is also the only angle given in thbe figure.

Therefore, The measure of m<B is 125°.

#2: We can calculate the value of the angles by means of the law of sine which is the following:

[tex] \frac{yz}{sin \: x} = \frac{xz}{sin \: y} = \frac{xy}{sin \: z} [/tex]

We need you know the Y value, therefore we replace and solve for Y

[tex] \frac{xz}{sin \: y} = \frac{xy}{sin \: z} \\ \frac{5}{sin \: y} = \frac{11}{sin \: 88} \\ 5 \: . \: sin \: 88 = 11 \: . \: sin \: y \\ sin \: y = \frac{5 \: . \: sin \: 88}{11} \\ sin \: y = 0.45426 \\ y = {sin}^{ - 1} (0.45426) \\ y = 27[/tex]

#3: In order to find the length of Df, we can use the law of sines in this triangle:

[tex] \frac{11}{sin \: (103)} = \frac{df}{sin \: (32)} \\ \frac{11}{0.974} = \frac{df}{0.53} \\ df = \frac{11.0.53}{0.974} \\ df = 5.99[/tex]

So the length of DF is 5.99 km.

#4: We are given two two angles and one side length.

<A = 37°

<B = 65°

AB = 13 ft

We are asked to find side length AC

We can use the "law of sines" to find the side length AC

[tex] \frac{sin \: c}{ab} = \frac{sin \: b}{ac} [/tex]

Let us first find the angle <C

Recall that the sum of all three interior angles of a triangle must be equal to 180°

[tex] < a + < b + < c = 180 \\ 37 + 65 + < c = 180 \\ 102 + < c = 180 \\ < c = 180 - 102 \\ < c = 78[/tex]

So, the angle <C is 78°

Now let us substitute all the known values into the law of sines formula and solve for AC

[tex] \frac{sin \: c}{ab} = \frac{sin \: b}{ac} \\ \frac{sin \: 75}{13} = \frac{sin \: 65}{ac} \\ ac = \frac{sin \: 65 \: . \: 13}{sin \: 75} \\ ac = \frac{9.06 \: . \: 13}{0.899} \\ ac = 13.1ft[/tex]

Therefore, the side length AC is 13.1 feet.

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