Algebra and Trigonometry: Further Applications of Trigonometry flashcards

Question 1

What is an oblique triangle?

Accepted Answer

A triangle that is not a right triangle.

Question 2

What does the Law of Sines state for a triangle with sides a, b, c opposite angles α, β, γ?

Accepted Answer

sin α / a = sin β / b = sin γ / c

Question 3

What is the ambiguous case when solving oblique triangles?

Accepted Answer

An SSA setup where two different triangles can both fit the given data.

Question 4

What are the possible numbers of triangles in the SSA case?

Accepted Answer

One, two, or none.

Question 5

What is the area of an oblique triangle with sides b and c and included angle α?

Accepted Answer

Area = (1/2)bc sin α

Question 6

What does the Law of Cosines give for a^2?

Accepted Answer

a^2 = b^2 + c^2 − 2bc cos α

Question 7

What is Heron's formula for the area of a triangle with sides a, b, c?

Accepted Answer

Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2

Question 8

What must be known to use Heron's formula?

Accepted Answer

All three side lengths.

Question 9

What is the pole of the polar grid?

Accepted Answer

Its origin.

Question 10

How do you convert polar (r, θ) to rectangular x and y?

Accepted Answer

x = r cos θ and y = r sin θ

Question 11

How are r, x and y related in polar conversion?

Accepted Answer

r^2 = x^2 + y^2

Question 12

For r = a ± b cos θ with a > 0 and b > 0, what curve results when a/b = 1?

Accepted Answer

A cardioid.

Question 13

For r = a ± b cos θ with a > 0 and b > 0, what curve results when a < b?

Accepted Answer

An inner-loop limaçon.

Question 14

What ratio a/b makes r = a ± b sin θ a dimpled limaçon?

Accepted Answer

1 < a/b < 2

Question 15

Which polar equations give a lemniscate?

Accepted Answer

r^2 = a^2 cos 2θ and r^2 = a^2 sin 2θ, with a ≠ 0

Question 16

How many petals does the rose curve r = a cos nθ (a ≠ 0) have?

Accepted Answer

2n petals if n is even; n petals if n is odd.

Question 17

What polar equation gives Archimedes' spiral?

Accepted Answer

r = θ, for θ ≥ 0

Question 18

What is the modulus |z| of the complex number z = a + bi?

Accepted Answer

|z| = √(a^2 + b^2), its distance from the origin.

Question 19

How is a complex number written in polar form?

Accepted Answer

z = r(cos θ + i sin θ)

Question 20

How do you multiply two complex numbers in polar form?

Accepted Answer

Multiply their moduli and add their angles.

Question 21

How do you divide two complex numbers in polar form?

Accepted Answer

Divide their moduli and subtract their angles.

Question 22

By De Moivre's Theorem, how is z^n found for a positive integer n?

Accepted Answer

Raise the modulus to the nth power and multiply the angle by n.

Question 23

What is the magnitude of the vector ⟨a, b⟩?

Accepted Answer

|v| = √(a^2 + b^2)

Question 24

How do you find the unit vector in the direction of a nonzero vector?

Accepted Answer

Divide the vector by its magnitude.

Question 25

What are the unit vectors i and j?

Accepted Answer

i = ⟨1, 0⟩ and j = ⟨0, 1⟩

Question 26

What is the dot product of two vectors in two dimensions?

Accepted Answer

The product of the horizontal components plus the product of the vertical components.

Algebra and Trigonometry: Further Applications of Trigonometry flashcards

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