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

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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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
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 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 Eulers formula can
be used to derive this equation. Write a short paragraph summarizing your work.

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