A limerick is a type of poem, usually humorous, that has 5 lines, in which the first, second, and fifth lines rhyme together and are longer, and the third and fourth lines rhyme together and are shorter. Here's an example:
A tall yet unassuming human being,
Wondered just what he was seeing,
His neighbor aghast,
Ran away really fast,
Shouting, "Don't look around while you're peeing!"
Can you come up with a Limerick that matches this mathematical equation??
((12 + 144 + 20 + 3*sqrt(4))/7)+(5*11) = 9^2 + 0
The solution is at the bottom of this page.
In the high school curriculum for Trigonometry, there is a lot of focus placed on two specific "special" right triangles: 30°-60°-90° and 45°-45°-90°. These triangles are deemed "special" because they have a fairly intuitive relationship between their side lengths. We can come up with a generalization of the relationship between the side lengths of each by assigning variables. For a 30°-60°-90° triangle, the lengths of the sides opposite the angles are x, x*sqrt(3), 2x, respectively. For a 45°-45°-90° triangle, the side lengths are x, x, x*sqrt(2).
We can then place these triangles within the unit circle in a standard orientation, with one side co-linear with the x-axis and the hypotenuse at a constant length of 1, such that the non-right angle vertex of the triangle is corresponds to a point on the unit circle. We can then label those points with exact coordinates based on our understanding of the different side lengths of these right triangles.
Our history classes tell the tale of Lewis and Clark's famous Native American guide, SOH CAH TOA, but neglect her important contributions to the field of mathematics. jklol. Moving beyond the x and y coordinates we are able to generate by placing triangles within the unit circle, trigonometric ratios allow us to discuss the general relationship between the side of a right triangle with respect to an angle. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite to the angle in a right triangle to the length of the side adjacent to the angle. If we consider that the hypotenuse of any triangle drawn within the unit circle is 1, that leads us to the statement that, within a right triangle, the sine of an angle is equal to the y coordinate, the cosine of an angle is equal to the x coordinate, and the tangent of an angle is equal to the y value divided by the x value.
When we define a trig ratio in this way, we notice some significant limitations. Specifically, they only apply to right triangles, and it seems as though we can only find the trig ratios with respect to acute angles. Examining the periodic and symmetric nature of a circle leads us to the discovery of what are called reference angles. Reference angles allow us to evaluate the trig functions for angles greater than 90°. Sine, cosine, and tangent values repeat over constant intervals around the unit circle. A 45° angle will have the same values for sine a cosine as a 135° angle (a 45° reference angle in quadrant 2), with the exception that the cosine of 135° will be -sqrt(2)/2 because x values are negative in quadrant 2. Reference angles are always drawn from the horizontal axis.
Applying this idea to all four quadrants of the unit circle allow us to generate values for sine, cosine, and tangent of every increment of 30° and 45° angles from 0° to 360°.
To find the trig ratios with respect to any angle around the unit circle, the main thing we need to be familiar with is the information we originally generated for quadrant 1, since this information just repeats with slight modifications as we move around the unit circle. When presented with an angle, one can sketch the location of the angle, identify which reference angle applies, write the corresponding coordinates, and specify the x value for cosine, the y value for sine, or the quotient of the y value divided by the x value for tangent, depending on what trig ratio was asked.
There are five other things we need to consider about the unit circle before we will be done discussing it:
We will do this in the coming weeks.
So that I can get a sense of how well you understand the material presented this week, please fill out this google form survey by 11:59PM on Sunday, April 12th.
A dozen, a gross, and a score,
Plus three times the square root of four,
Divided by seven,
Plus five times eleven,
Is nine squared and not any more.