“Vector Components”

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Laboratory: Basic Trigonometry and Vector Components Goals: Explore trigonometric functions Sine and Cosine and the components

of a 2D Vector.

Requirements: Please read the section(s) in your text on Vectors and review (if included) the Appendix on Trigonometric Operations.

Background: In this activity we will understand the coordinate points on the

circumference of the unit circle and the values of the function’s sin() and cos(). With this information, we will understand how the sin() and cos() are used along with an angle to determine the components of a 2D vector.

Note: This lab has 8 questions. n Click the link below to render the virutal lab

https://phet.colorado.edu/sims/html/trig-tour/latest/trig-tour_en.html You should see this:

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Exercises:

1. Understanding the Unit Circle In this activity we study the unit circle, angles, sin(), cos(), and components.

a. Setup i. Select (to the right) cos ii. Select Special angles iii. Select Labels

b. Your lab should look like this:

c. For reference, the circle is divided into 4 Quadrants, as depicted below. We will reference the quadrants throughout the lab.

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d. Question # 1: What is the radius of the circle? e. Question # 2: Why is it called a unit circle? f. Question # 3: Along the circumference of the circle are tiny circles

representing points for ‘Special Angles.” What are the angles (starting with 0º) in the quadrants I and II? (Click and hold the red dot to move around the circle. The left box titled Values shows the angle)

g. Some background: Below is a basic diagram showing the unit circle and the definitions of sin(θ) and cos(θ).

In our case, r, the length of the ‘hypotenuse’ is equal to 1 (r = 1). Rearranging the equations above, we see that

The triangle above defined by x, y, and r, is called a Right Triangle because the sides x and y are at a right angle (90º) to each other. For a right triangle, x, y, and r are related by the Pythagorean Theorem (named after a Greek mathematician who discoed it):

It follows that

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And with r = 1

Squaring both sides, we see for our circle or radius 1

2. Exercises a. In the lab, construct a right triangle at 60º (in quadrant I).

i. Question #4: What are the (x, y) values? ii. Question #5: Using the Pythagorean Theorem and the (x, y)

values from Question #4, what is the length of the hypotenuse. The length of the hypotenuse can be considered the magnitude of a vector (tail at the origin and head on the unit circle), and the angle the vector makes with the x-coordinate can be considered the direction the vector is moving. The (x, y) values can be considered the magnitudes of the component vectors moving in the horizontal (x axis) and vertical (y axis).

b. In the lab, construct a vector in quadrant III at 240º. i. Question #6: What are the (x, y) values or the magnitudes of the x

and y component vectors, respectively? Adding vectors is nothing more than adding all x components of each vector to get the resultant x component and adding all y components of all vectors to get the resultant y component (don’t mix x’s and y’s). See the diagram below.

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ii. Question #7: What are the resultant x and y components of adding the vectors from Question #6 and Question #4?

iii. Question #8: Use the Pythagorean Theorem to find the magnitude of the resultant vector.