Portfolio Skills 2 and 4 -- Nora
- Author:
- Nora Teter
P2. I can explain why the sine ratio and the cosine ratio are the (x, y) coordinates on a unit circle.
*the cos and sin values of angle a can be found in the left side bar under the subtitle “function”.
This visual explains both why the sine ratio and the cosine ratio are the (x, y) coordinates on a unit circle. SOH CAH TOA (sin= opposite/hypotenuse, cos=adjacent/hypotenuse, tan=opposite/adjacent) is also something to keep in mind in this explanation.
A unit circle always has a radius of one (hence the “unit”), therefor the right angle triangle in a unit circle will always have a hypotenuse of 1. Since sin=opposite/hypotenuse and cos=adjacent/hypotenuse, and since hypotenuse=1, in a unit circle, the values of sin and cos will actually just be the values of opposite and adjacent, respectively.
Additionally, another interesting feature is that the opposite side of the triangle (the side that correlates with sin) is parallel to the y axis (the axis that correlates with cos). Likewise, the adjacent side of the triangle (the side that correlates with cos) is parallel to the x axis (the axis that correlates with cos).
Essentially, not only does the numerator in a sin function actually equal the quotient of the function, but both the quotient and numerator represent a line that is parallel to the y axis. Similarly, this exact scenario can be applied to a cos function, with the line/quotient/numerator parallel to the x axis.
Thus, the fact that the numerator in the equation for sin/the quotient of a sin function involves a line that is parallel to the y axis, shows how the value of sin is really just the equivalent of the y coordinate on the circle. Likewise, the fact that the numerator in the equation for cos/the quotient of a cos function involves a line that is parallel to the x axis, shows how the value of cos is really just the equivalent of the x coordinate on the circle.
This can also be demonstrated by dragging the point around the circle, and seeing how the y coordinate on the circle, the length of the opposite side of the triangle, and the sin value are all equal. Likewise, the x coordinate on the circle, the length of the adjacent side of the triangle, and the cos value are all equal.
While working on this visual, I discovered some more interesting properties relating to the x-y coordinates and the cos and sin values:
As you drag point D around the unit circle, you can see how the location of the point (relative to x and y) correlates with the cos and sin values of the angle measurement. Depending on which axis the terminating side of the angle is closer to, the absolute value of the cos and sin functions will be increase or decrease relative to the axis that it correlates with. If the terminating side of the angle is closer to the x axis, the absolute value of cos will be larger than the absolute value of sin, if it's closer to the y axis the absolute sin value will be larger than the absolute cos value.
For example:
If you start with point D at (0.92,0.4), the cos value (.92/1=.92 , |.92|=.92) is greater than than the sin value (.4/1=|.4|=.4) because point D is closer to the x axis. Cos correlates with the x axis, so it is the greater of the values when the terminating side of the angle is closer to the x axis.
If you start with point D at (-0.27,-0.96), the sin value (-0.96/1=-0.96, |-.92|=.92) is greater than the absolute cos value (-.27/1=|-.27|=.27) because point D is closer to the y axis. Sin correlates with the y axis, so it is the lesser of the values.
P4: I can explain how right triangle trigonometry relates to unit circle trigonometry.
*disclaimer: when discussing values of cos and sin and x-y coordinates in this section, please assume all values to be the absolute values of themselves. Sin and cos values with vary in whether they are positive or negative by which quadrant they’re located in, however the lengths of the sides of the triangle will always remain positive. Therefor, any and all statements which claim all cos and sin values to be equal to the side lengths are referring to the absolute value of those previously stated cos and sin values.
Unit circle trigonometry is essentially just an extension of right triangle trigonometry, as it involves right triangles but integrates circles and it takes place on an x-y plane. Right triangle trig acts as a sort of introduction to unit circle trig, as it familiarizes with triangles, but also introduces sin, cos, and tan. Unit Circle trig acts as an introduction to analytical trig, which is essentially an application of trig that falls outside of the conventional high school math norm. Analytical trig is not about solving triangles, as it usually is used in solving problems focused on calculus or physics. Unit circle trig serves as an introduction to this, as it precedes physics oriented math (like analyzing waves) or calculus oriented math (like analyzing the patterns in functions). Unit circle trig goes on to analyze things that rotate or vibrate, such as light, sound, or orbit; more applicable concepts that involve things like sinusoidal waves. Essentially, unit circle trigonometry allows for the maximum capacity of trigonometry, while right angle trig is the more limited, introduction form of the former.
In unit circles, right angle triangles are commonly used for measuring angles, solving functions, and determining point values. As you can see in the visual, no matter where point “D” is on the circle, the triangle will remain a right angle triangle that is based on the x axis. The only times that a point on a circle does not create a right triangle is when the point rests directly on either x or y axis.
In order to determine the coordinates of any given point on a circle, one must look to the reference angle within the circle. The reference angle, which is always within the circles right triangle, is a connection of a point on the x-axis to the origin to a point on the unit circle.
When point D is in Q1, the right triangle is both the angle measurement and the reference angle. The equation: 0+reference angle=angle measurement
When point D is in Q2, the right triangle is the reference angle for the angle that exceeds 90˚. The equation: 180-reference angle=angle measurement
When point D is in Q3, the right triangle is the reference angle for the angle that exceeds 180˚. The equation: 180+reference angle=angle measurement
When point D is in Q4, the right triangle is the reference angle for the angle that exceeds 270˚. The equation: 360-reference angle=angle measurement
Essentially, the unit circle uses the right triangle as a vessel to determine the point on the circle by utilizing the measurement of the reference angle, or vice versa.
This “vessel” or right triangle is made up of the radius of the unit circle, a line parallel to the x axis, and a line parallel to the y axis.