During this unit, we learned about eight different trigonometry formulas. The formulas we learned about were sine, cosine, tangent, ArcSine, ArcCosine, ArcTangent, Law of Sines and the Law of Cosines. These formulas are very important, especially when it comes to finding the angles and sides of triangles.
We weren't directly introduced to all of these concepts at once. We slowly began to ease ourselves in to new concepts and formulas for working with triangles. The first thing that we began this unit with was proving the Pythagorean Theorem. The most basic definition of the Pythagorean Theorem is: a squared + b squared = c squared. We had to learn to use and prove this formula, using a basic problem called 'proof by rugs'. By slowly recognizing the relationship between the shapes, we slowly moved on to more complex problems. The second problem was when we actually began splitting up triangles and using the theorem as a formula rather than just seeing the relationships between the triangles right in front of us. Then, we began to learn how the Pythagorean Theorem correlates to the distance formula. While the Pythagorean Theorem is a squared + b squared = c squared, the distance formula is D=√(x1−x2)2+(y1−y2)2. One of the biggest similarities to the two is the different situations they're used in. While the distance formula is used using points, the Pythagorean Theorem is almost the same thing in a different context. An important thing to know about for going forward after the distance formula and the Pythagorean Theorem, is the unit circle. Knowing an understanding the unit circle is a very important thing when it comes to trigonometry. Although, when you first find the points on the circle, youĺl be able to use the distance formula (described in the paragraph above) to plug in the points on the circle. To identify those points on a unit circle, you have to draw line segments and create angles. Youĺl also end up creating the angle theta, which will come in handy when you move into sines and cosines. Once you have the correct points in the circle, you can draw and label a triangle inside the unit circle (using the Pythagorean Theorem) to find the side/angle of the circle. You can also use symmetry to find the missing points on the unit circle. By reflecting the radius line across the x-axis, we can get symmetrical angles to help us find the missing points sooner. Then, if we reflect the two radius lines across the Y axis, we can get two other angles/sides. Now i'm going to move on to sine and cosine. Something that's given is that the cosine of theta is x, and the sine of theta is y. Cosine and Sine are two pretty important things you need to know in Trig. Using sine and cosine, we can actually find side lengths using those functions on the calclulator. This is a key factor in finding Skipping around a bit, another important thing to know for this unit is the tangent. Tangent, is written T=O/A. You can use Tangent to help you find a specific point on a circle, just like Sine and Cosine. (Defined in the paragraph above.) Cosine, Tangent, and Sine all become very important when working with arcCosine, arcTangent, and arcSine, especially when working with the unit circle. "arc" is basically the opposite of each of the formulas. These formulas are called "inverse trigonometry functions". In class, we used a problem involving Mount In Everest to help us introduce the concepts of the Law of Sines; The Law of Sines is a formula that helps us find either the side or angles of a triangle, it depends on how you use it. Using this formula, you can either find the side lengths or angles of a triangle, it just depends on how you substitute your known numbers into the equation. The Mount Everest problem we did in class forces us to use this equation in one way or another. For that problem I had to use one of my favorite Habits of a Math Matician, which is starting small. I have to use that habit on a lot of the problems in this unit, especially when it comes to the Unit Circle.
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