8 Trigonometric Formulas
Sine:
The trigonometric function this is equal to the ratio of the side of a given angle in a right triangle to the hypotenuse.
Cosine:
The trigonometric function that is equal to the ratio of the side adjacent to an acute angle in a right triangle to the hypotenuse.
Tangent:
A straight line or plane that touches a curve/curved surface at a point, but if extended does not cross it at that point
Arcsine:
A mathematical function that is the inverse (opposite) of the sine function.
ArcCosine:
A mathematical function that is the inverse (opposite) of the cosine function.
Arctangent:
A mathematical function that is the inverse (opposite) of the tangent function.
Law of sines:
The law of sines provides a formula that relates the sides with the angles of a triangle. The formula allows you to find the side length of the angle of any triangle.
Law of cosines:
The law of cosines helps you for calculating one side of a triangle when the angle opposite and the other two sides are known.
The trigonometric function this is equal to the ratio of the side of a given angle in a right triangle to the hypotenuse.
Cosine:
The trigonometric function that is equal to the ratio of the side adjacent to an acute angle in a right triangle to the hypotenuse.
Tangent:
A straight line or plane that touches a curve/curved surface at a point, but if extended does not cross it at that point
Arcsine:
A mathematical function that is the inverse (opposite) of the sine function.
ArcCosine:
A mathematical function that is the inverse (opposite) of the cosine function.
Arctangent:
A mathematical function that is the inverse (opposite) of the tangent function.
Law of sines:
The law of sines provides a formula that relates the sides with the angles of a triangle. The formula allows you to find the side length of the angle of any triangle.
Law of cosines:
The law of cosines helps you for calculating one side of a triangle when the angle opposite and the other two sides are known.
Measuring your World Narrative
Something we did at the beginning of this unit was that we re-created a proof to prove the Pythagorean Theorem. The proof involved proving triangles were similar, setting up proportions based on similarity, writing equations, and solving the equations to show that a^{2}+b^{2}=c^{2}. The Pythagorean Theorem was later useful when we derived the distance formulad=(x1-x2)2+(y1-y2)2. This is because if you have two points on a graph you can just form of right triangle because the two points meet at a vertices. You can find those missing lengths by using Pythagorean Theorem, because you have a right triangle. When we started working with distance formula we just reviewed how to use the formula d=(x1-x2)2+(y1-y2)2 and how to solve the problems by plugging in the numbers. This was laters useful when we started working with the equations of a circle. This is because the equations of a circle is somewhat similar to the distance formula. The formula is x2+y2=r2, so you can see it is similar to what we were working with before, and we were using it to derive an equation of a circle centered at the origin of a Cartesian coordinate plane.The unit circle is a circle with a radius of one and center of O and a starting origin of (0,0). We used the unit circle to understand the trigonometric angle measures by forming right triangles in the unit circle. We learned about the unit circle by finding two coordinate points of a line at 30 degrees, 45 degrees and 60 degrees. We found the coordinate points by using Pythagorean Theorem. That led to using the symmetry of the circle to find the rest of the points on the unit circle, we did this by dropping a perpendicular to make a right angle triangle. After we completed the unit circle we were able to define sine and cosine of the angle theta (θ). We started learning about sine and cosine by doing simple trigonometry practice (Cosine = adjacent/hypotenuse and Sine= opposite/hypotenuses) we found the missing side lengths. Also we were able to check our trig work by using the pythagorean theorem, which we had learn at the begging of the unit. Then we learned about the tangent function by doing a proof, and we learned a tangent is a single point, and a tangent and a radio line meet at 90 degrees (Cos θ=X & Sin θ Y, & Tan θ= Y/X). When we started to use similarity and proportions to derive the general trigonometric functions (sine, cosine, and tangent), we were to set up a right triangle and set up proportions when proving a formula. We used the unit circle to define the arcSine, arcCosine and arcTangent functions by looking at how the different points allowed for us to solve for angles in a triangle. Towards the end of the unit we received the Mount Everest problem to discover the Law of Sines. We learned about the law of sines by taking it apart, and solving the Mount Everest problem. Then after that problem we went a little deeper in learning about the law of sines by doing a few practice problems and establishing that law of sines is angle, side, angle). While we were learning about the law of sines, we got introduced to the law of cosines (SAS). Overall the law of cosines is just a transformed version of the pythagorean theorem that is used to find the side lengths when you have side, angle, side. Overall the pythagorean theorem played probably the most important role in this unit because it was always involved in the process of our learning because it ether always went back to the formula or the formula was rearranged to help us solve other problems.