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## AP®︎/College Physics 1

### Course: AP®︎/College Physics 1>Unit 2

Lesson 2: Analyzing vectors using trigonometry

# Analyzing vectors using trigonometry review

Review the skills for analyzing vectors, including how to find horizontal and vertical components of vectors.

## Analyzing vectors with trigonometry

To simplify calculations for two-dimensional motion, we analyze the movement in the vertical direction separately from the horizontal direction. Since displacement, velocity, and acceleration are vector quantities, we can analyze the horizontal and vertical components of each using some trigonometry.

### Finding horizontal and vertical components

We can find the horizontal component A, start subscript, x, end subscript and vertical component A, start subscript, y, end subscript of a vector using the following relationships for a right triangle (see Figure 1a). A is the hypotenuse of the right triangle.
A, start subscript, x, end subscript, equals, A, cosine, theta
A, start subscript, y, end subscript, equals, A, sine, theta
Figure 1a: We analyze a vector by breaking it down into its perpendicular components, A, start subscript, x, end subscript and A, start subscript, y, end subscript.

### Determining the magnitude of the resultant

When we know the horizontal and vertical components, we can find the magnitude of their sum using the Pythagorean theorem (Figure 2).
open vertical bar, A, close vertical bar, equals, square root of, A, start subscript, x, end subscript, squared, plus, A, start subscript, y, end subscript, squared, end square root
Figure 2: Given the horizontal component, A, start subscript, x, end subscript, and vertical component, A, start subscript, y, end subscript, we can find the magnitude of the vector sum A and angle theta.

### Finding vector direction

To find the angle theta of the vector from the horizontal axis, we can use the horizontal component A, start subscript, x, end subscript and vertical component A, start subscript, y, end subscript in the trigonometric identity:
tangent, theta, equals, open vertical bar, start fraction, A, start subscript, y, end subscript, divided by, A, start subscript, x, end subscript, end fraction, close vertical bar
We take the inverse of the tangent function to find the angle theta:
theta, equals, tangent, start superscript, minus, 1, end superscript, open vertical bar, start fraction, A, start subscript, y, end subscript, divided by, A, start subscript, x, end subscript, end fraction, close vertical bar

## Common mistakes and misconceptions

Sometimes people forget when to use sine or cosine for calculating vector components. When in doubt, draw a right triangle and remember:
\begin{aligned}\sin \theta &=\dfrac{\text {opposite}}{\text{hypoteneuse}}\\ \\ \cos \theta&=\dfrac{\text{adjacent}}{\text{hypoteneuse}}\\ \\ \tan \theta &=\dfrac{\text{opposite}}{\text{adjacent}}\end{aligned}