Jamal draws triangle XYZ. If angle X=46 degrees, XZ = 10 degrees, and YZ=8 degrees, what is the approximate length of XY?
2 answers:
<span>In triangle XYZ,
angle X=46 degrees,
XZ = 10 units and
YZ=8 units.
We will find the length of XY using cosine rule:
According to cosine rule:
a² = b² + c² - 2bc cosA (the figure is attached)
Here, A = 46 degree.
a = 8 units.
b <span>= ?
</span>c = 10 units.
</span>⇒ b² = c² - a² -2bc cosA
⇒ b² = 100 - 64 - 160*cos(46)
⇒ b² = 100 - 64 - 160*0.69
⇒ b² = 100 - 64 - 110.4
⇒ b² = 100 - 64 - 110.4
⇒ b² = 74.4
or b = 8.62 units.
or b = 8 untis approximately.
angle X=46 degrees,
XZ = 10 units and
YZ=8 units.
We will find the length of XY using cosine rule:
According to cosine rule:
a² = b² + c² - 2bc cosA (the figure is attached)
Here, A = 46 degree.
a = 8 units.
b <span>= ?
</span>c = 10 units.
</span>⇒ b² = c² - a² -2bc cosA
⇒ b² = 100 - 64 - 160*cos(46)
⇒ b² = 100 - 64 - 160*0.69
⇒ b² = 100 - 64 - 110.4
⇒ b² = 100 - 64 - 110.4
⇒ b² = 74.4
or b = 8.62 units.
or b = 8 untis approximately.
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Answer:
distance: 320.624 miles
direction: 43.576°
Step-by-step explanation:
The speed and direction can be found by adding the given vectors.
... 320∠40° + 20∠130°
... = (320·cos(40°), 320·sin(40°)) + (20·cos(130°), 20·sin(130°))
... = (245.134, 205.692) +(-12.856, 15.321) = (232.278, 221.013)
The magnitude of the vector with these components is found using the Pythagorean theorem. The direction is found using the arctangent function.
... = √(232.278² +221.013²)∠arctan(221.013/232.278)
... = 320.624∠43.576°
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A suitable vector or graphing calculator can do this easily. In the screenshot of a TI-84 app below, the variable D has the value π/180. The display mode is set to degrees.
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<em>Comment on coordinate systems</em>
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Reflection does not change lengths or angles within a given geometry. Hence, we can use all the usual tools of vector calculation using navigation coordinates, without bothering to translate them back and forth to x-y coordinates.
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Problems like this generally can be worked using the Law of Cosines and the Law of Sines, too. It generally helps to draw a diagram so you can find the values of the angles betwee the various vectors more easily.
