Tutor profile: Andrea L.
How do you write Cosine of an angle?
First, it would be helpful to remember the acronym SOH-CAH-TOA ((A stands for adjacent, H stands for hypotenuse, O stands for opposite). "CAH" tells us that cosine of an angle is the ratio of the "Adjacent side to the Hypotenuse" in a right triangle. When we label our triangles sides we always label from the angle we are writing the ratio of. The "opposite" side is the one across the triangle of our angle. The "hypotenuse" is the side across from the right angle and the "adjacent" side is the side next to the angle that is NOT the hypotenuse. A ratio is just a comparison of two quantities. Often we write these as fractions. If the "adjacent" side of our triangle is 12 units and the "hypotenuse" is 15 units, then cosine (CAH in our acronym) would be 12/15. Please let me know if you have any other questions!
How can you prove a figure is a rectangle in the coordinate plane?
A rectangle is a special parallelogram. (Think like a husky is a special kind of dog) There are multiple ways you can go about proving this scenario. One of these is to first show the figure is a parallelogram (think a dog as far as the other scenario- the broader category). Using the definition of a parallelogram, it is a four sided figure with two pairs of parallel sides. We could compare slopes of opposite sides to determine slopes of opposite sides are equal (meaning the segments are parallel). Now to show that the figure is a more specific parallelogram- a rectangle (think a husky in the animal scenario- a more specific kind of dog)- we have to show at least one unique property of a rectangle is true for the figure. We have the following two options: 1. We could show that there is one right angle. We could again use slopes of sides that are right next to each other. This time we want to show they are perpendicular (opposite reciprocals). 2. We could also use a different route of showing that the diagonals of the figure are congruent (equal in length). This would involve having to use the distance formula (d=sqrt((x2-x1)^2+(y2-y1)^2). This would be more work, but would definitely be another option! Please let me know if you have any other questions!
Remember that the symbols "|" mean distance from 0. When we think about distance it is possible to move 0 feet, 2 feet, 5 feet. It is NOT possible to move -5 feet. With this in mind, another way to ask the question above is "where on a number line is "x+3" 5 units away from zero"? We could be 5 units away (in either direction). This results in two separate scenarios. x+3 = 5 (we could be located at 5 on the number line as it is 5 units from 0) or the second scenario is x+3 = -5 (we could be located at -5 on the number line as it is also 5 units from 0). Solving both of these equations by subtracting 3 on both sides would result in two solutions of 2 and -8. Please let me know if you have any other questions!
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