1. State the following: a. State that “Properties of Absolute Values (b > 0)” b. State that “The Pythagorean Identities” c. State that “Horizontal Line Test” d. State that “The Law of Sines” e. State that “The Law of Cosines”
1. State the following:
a. State that “Properties of Absolute Values (b > 0)”
b. State that “The Pythagorean Identities”
c. State that “Horizontal Line Test”
d. State that “The Law of Sines”
e. State that “The Law of Cosines”
![](/static/compass_v2/shared-icons/check-mark.png)
a. “Properties of Absolute Values (b > 0)”
- Non-negativity |a| ≥ 0
- Positive-definiteness |a| = 0a = 0
- Multiplicativity |ab| = |a| |b|
- Subadditivity |a + b| ≤ |a| + |b|
- Idempotence ||a|| = |a|
- Symmetry |−a| = |a|
- Identity of indiscernible |a − b| = 0 ⇔ a = b
- Triangle inequality |a − b| ≤ |a − c| + |c − b|
- Preservation of division |a/b|=|a|/|b| if b ≠ 0
b. “The Pythagorean Identities”
When studying the unit circle (radius of 1), it was observed that a point on the unit circle (a vertex of the right triangle) can be represented by the coordinates (cos θ, sin θ ).
Since the legs of the right triangle in the unit circle have the values of sin θ and cos θ, the Pythagorean Theorem can be used to obtain sin2 θ + cos2 θ = 1.
This well-known equation is called a Pythagorean Identity.
It is true for all values of θ in the unit circle.
Using this first Pythagorean Identity, two additional Pythagorean Identities can be created.
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