Math  /  Trigonometry

QuestionSelect all statements that are true. θ\theta is in standard position. (a,b)(a, b) are the coordinates of a point on the terminal arm of θ\theta If you know the values of a and bb, you can determine the value of θ\theta If you know the value of θ\theta, you can determine possible values for aa and bb.

Studdy Solution

STEP 1

What is this asking? Which statements are true about the relationship between an angle θ\theta and a point (a,b)(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 aa and bb, can we find θ\theta?
Imagine a point (a,b)(a, b) plotted on a coordinate plane.
The angle θ\theta is formed by the positive x-axis and a line drawn from the origin to this point.

STEP 4

Let's say a=1a = 1 and b=1b = 1.
We can visualize this point.
The angle θ\theta would be 4545^\circ or π4\frac{\pi}{4} radians.

STEP 5

Now, let's say a=1a = -1 and b=1b = 1.
This is a different point!
The angle θ\theta would be 135135^\circ or 3π4\frac{3\pi}{4} radians.
Knowing aa and bb *does* let us find θ\theta!
So the first statement is **TRUE**.

STEP 6

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

STEP 7

Let's say θ=45\theta = 45^\circ or π4\frac{\pi}{4} radians.
The point (1,1)(1, 1) is on the terminal arm.
But so is (2,2)(2, 2), and (3,3)(3, 3), and infinitely many other points!

STEP 8

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

STEP 9

Both the first and second statements are **TRUE**.

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