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

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

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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 T, 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 Eulers formula, after the Swiss mathematician
Leonhard Euler (1707-1783).
ea + bi = ea(cos b + i sin b)

This formula gives rise to the equation epi + 1 = 0. This equation related the five most famous

numbers in mathematics–0, 1,p , e, and i– in a single equation. Show how Eulers formula can

be used to derive this equation. Write a short paragraph summarizing your work.

…

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