Solving the right triangle
- 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.
- 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.
- a) In your own words, state the difference between a scalar and a vector. Give examples of each. b) Give geometric descriptions of the operations of addition of vectors and multiplication of a vector by a scalar.
4. What is known about Θ, the angle between two nonzero vectors u and v, if each of the following is true? Explain your answers.
a) u • v = 0
b) u • v > 0
c) u • v < 0
5. The famous formula shown below is called Euler’s formula, after the Swiss mathematician Leonhard Euler (1707-1783).
e^a + bi = e^a(cos b + i sin b)
This formula gives rise to the equation e^πi + 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.
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