Math Snap

STEP 1

What is this asking?
Which statements are true about the relationship between an angle $\theta$ and a point $(a, b)$ on its terminal arm?
Watch out!
Don't mix up which values determine which other values!

STEP 2

1. Analyze the first statement
2. Analyze the second statement

STEP 3

If we know $a$ and $b$, can we find $\theta$?
Imagine a point $(a, b)$ plotted on a coordinate plane.
The angle $\theta$ is formed by the positive x-axis and a line drawn from the origin to this point.

STEP 4

Let's say $a = 1$ and $b = 1$.
We can visualize this point.
The angle $\theta$ would be $45^\circ$ or $\frac{\pi}{4}$ radians.

STEP 5

Now, let's say $a = -1$ and $b = 1$.
This is a different point!
The angle $\theta$ would be $135^\circ$ or $\frac{3\pi}{4}$ radians.
Knowing $a$ and $b$ does let us find $\theta$!
So the first statement is TRUE.

STEP 6

If we know $\theta$, can we find $a$ and $b$?
Imagine an angle $\theta$ on a coordinate plane.
There are many points $(a, b)$ that lie on the terminal arm of this angle!

STEP 7

Let's say $\theta = 45^\circ$ or $\frac{\pi}{4}$ radians.
The point $(1, 1)$ is on the terminal arm.
But so is $(2, 2)$, and $(3, 3)$, and infinitely many other points!

STEP 8

Knowing $\theta$ doesn't give us specific values for $a$ and $b$.
It tells us the ratio between $a$ and $b$, but not their exact values.
So, while we can find possible values, we can't determine the values.
The second statement is TRUE.

Both the first and second statements are TRUE.