Law of sines, law of cosines, vectors in the plane, vectors and dot products, trigonometric form of complex numbers

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Five questions: Law of sines, law of cosines, vectors in the plane, vectors and dot products, trigonometric form of complex numbers.

Question 1: Law of Sines Can the Law of Sines be used to solve a right triangle? Write a short paragraph explaining why it can or cannot be used. If it can be used, explain how to use it. Is there an easier way to solve the triangle? Explain. Question 2: Law of Cosines Describe how the Law of Cosines can be used to solve the ambiguous case of the oblique triangle ABC, where a = 12 feet, b = 30 feet, and A = 20°. Is the result the same as when the Law of Sines is used to solve the triangle? Describe the advantages and the disadvantages of each method. Question 3: Vectors in the Plane 1. In your own words, state the difference between a scalar and a vector. Give examples of each. 2. Give geometric descriptions of the operations of addition of vectors and multiplication of a vector by a scalar. Question 4: Vectors and Dot Products What is known about Θ, the angle between two nonzero vectors u and v, if each of the following is true? Explain your answers. 1. u • v = 0 2. u • v > 0 3. u • v < 0 Question 5: Trigonometric Form of Complex Numbers The famous formula shown below is called Euler’s formula, after the Swiss mathematician Leonhard Euler (1707-1783). ea + bi = ea(cos b + i sin b) This formula gives rise to the equation eπi + 1 = 0. This equation related the five most famous numbers in mathematics--0, 1,π , e, and i-- in a single equation. Show how Euler’s formula can be used to derive this equation. Write a short paragraph summarizing your work.

Tutor Answer

pallveechem123
School: Rice University

here is solution; kindly speak to me if any doubt you have .

Can the Law of Sines be used to solve a right triangle? Write a short paragraph explaining
why it can or cannot be used. If it can be used, explain how to use it. Is there an easier way
to solve the triangle? Explain.
A triangle has six entities, namely three sides and three angles. If one is given one of following
combination:
(i) Two sides and an angle,
(ii) One side and two angles
One can readily get the rest of entities using Law of sines.The Law of Sines says that in any
given triangle, the ratio of any side length to the sine of its opposite angle is the same for all
three sides of the triangle. This is true for any triangle, including for right triangles.
sin( A) sin( B) sin( C )


……… (i)
a
b
c
In triangle, let angles be C  90 ; A   and B  90   .

Substituting in equation (1), we have
sin(  ) sin( 9...

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Anonymous
Awesome! Exactly what I wanted.

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