Finding Exact Values of Sine, Cosine, and Tangent on a Unit Circle

The unit circle is a circle on the coordinate plane whose radius is "1." We like the number 1 as a radius because it's super-simple and, generally, the easiest non-zero whole number to work with. The unit circle looks like this:

and we began class by making a reference picture to fill in all the exact coordinates of the unit circle. Will we ever be tested on any angle measures BESIDES the ones on the unit circle? NO. These are the most common and the only ones that you need.

From prior knowledge, we know that sin = y, cos = x, and tan = y/x. So to find the exact trig values for a given angle in degrees or radians, simply insert your y value if you're finding sine or your x value if you're finding cosine. Tangent is a little more tricky: you have to simplify y/x and divide your fractions. More often than not, something will cancel out.

Review for Quiz

2nd Period: We reviewed for the quiz and you took it in the same period - well done! Most of you did really well and I'm proud of your efforts. Shout out to those who stayed after the bell to make sure you finished your work.

4th and 6th Periods: Due to my poor planning and lack of time due to the pep rally, our quiz will be on Monday. We made the cheat sheets in class today - don't lose them! There are 5 quiz questions and 1 bonus question. You will need to know:

1. How to calculate an arc of a circle
2. How to calculate the area of a sector
3. How to work a problem with two interlocking arcs moving together
4. How to distinguish between arc and area questions.

HINT for the Review Sheet due next Friday 10/4

For Questions 5-7:

The number on the outside of the circle represents the length of the arc. Notice that you aren't given the central angle: you're going to have to solve to find it. Start with the formula for the length of an arc. Plug in the arc length and the radius, and solve for theta. Conveniently, this puts theta in radians, which is what we want it to be in anyway. Then, plus in the theta and the radius back into the formula for the area of the sector, and find the area (putting the correct units on at the end!).

You can use this strategy for all 3 questions.

Applications of Sectors

We did a weird experiment today with sector measurements, protractors, string compasses, and oatmeal creme pies (sorry if you missed it!). We cut away pieces of the oatmeal creme pie and measured the remaining part of the sector using the compass, ruler, protractor, and the formula for the area of a compass.

Don't forget that when the problem asks you for the REMAINING area of the sector, they are wanting you to do 360 - the given angle = the angle you want to use. This will be your central angle for the remaining sector. Then convert it into radians and plug it into the formula for the area.

Area of a Sector

While arcs can be thought of as a perimeter of a circle, sectors can be thought of as the area of (part of) a circle. It's formula is very similar to that of the arc formula:

where r represents the radius and theta represents the degree of the central angle in RADIANS not degrees (if it is in degrees, then convert it).  Remember that your final answer must include units and, since we're finding the area of something, the units must be squared.

Applications of Trigonometric Arcs - Airplane Travel

Our activity today was based upon using arc lengths to calculate distances across the Earth. Given that the radius of Earth is approximately 4000 miles, we can calculate the distance between two cities if we know the central angle (converted to radians, of course) between those two cities.

This turned into a group effort where students found distances to fly their planes while trying to get paid more than any other group. Fuel efficiency was taken into consideration and there was a travel limit based on the number of gallons that the plane began with.

Shout out to Jay Rice, Kyle Krus, and Chris Coleman for developing an algorithm to determine the MAXIMUM amount of money that could possibly be made given the amount of gas in the tank, the fuel efficiency, and the most profitable destinations - WELL DONE!

Another shout out to the group of QuaDarius Gatewood, Joe Tripp, and James Petty for earning over $100,000 with their plane.

Applications of Arc Lengths

We know that to find the length of an arc, we must multiply the radius of the circle times the interior angle of the circle (or central angle) so long as the central angle is in radians, not degrees. In circular trigonometry, we always measure in RADIANS. Make sure to convert first!

Today, we found the missing angle of a larger/smaller circle given the radius/central angle of another. To do this, we first found the missing arc length, then set it equal to the arc length of the other circle to solve for the missing central angle.

For a real-life application of this problem, we looked at gears on a clock and how they rotated together. Depending on the size of the gear, the gear rotated a different amount of total degrees (which we solved using the s = r times theta formula).

Review for the Angles and Triangles Test

Your first test is tomorrow! Remember that you are allowed to bring a 1/2 page cheat sheet of notes - front side only. There are 10 questions and 2 bonus questions on your test. Make sure that you show all of your work for each question.

The things you should be able to know how to do for this test are:

1. Solve trig equations
2. Simplify trig reciprocal identities
3. Find the six trig functions on a graph given a point or a slope
4. Find the missing sides of a right triangle
5. Find the missing angles of a right triangle (and convert them to radians for bonus)
6. Simplify expressions using properties of quadrantal angles.

Good luck and happy studying!

Solving Trig Equations

Solving trig equations are just like solving regular equations. First, do any distributing and combining like terms that you see. Get your thetas with your thetas and your numbers with your numbers - try and keep theta positive if possible. Then, simplify your equation until you only have one number left, and it's next to a trig function. Perform the trig operation, then use the inverse functions on the calculator to solve for theta, the missing angle.

Remember that you always want to have the SAME trig function in your equation when you solve. If you don't have this, you may have to use a reciprocal identity to simplify your equation first before you begin to solve.

Reciprocal Identities

An identity in math is something that is equal after both sides of the equation are simplified. Reciprocal identities simply mean the opposite of each trig function. We have three fundamental reciprocal identities in trig:

sin = 1/csc,    cos = 1/sec,    tan = 1/cot

We can simplify expressions using reciprocal identities by first flipping the numerator and denominator. Then, check to see if the radical is in the numerator, if not, then multiply by the radical to get it into the numerator (and out of the denominator). Next, check and see if the radical is in simplest terms. If not, break it down and take out any pairs of numbers that you find (anything that is not a pair is multiplied back together and stays under the radical). Finally, check to make sure that your fraction is in simplest terms. If not, reduce it. Don't forget to include the trig function as part of your answer!

Signs and Ranges

Our six trig functions have different signs depending on their location on the coordinate plane. In quadrant I, all trig functions are positive. In Quadrant II, only sine and cosecant are positive. In Quandrant III, only tangent and cotangent are positive. And in Quadrant IV, only cosine and secant are positive. This holds true for all triangles that we can draw on a coordinate plane, and will come into greater effect later on in the year.

Guest Speaker from Fayetteville/Mr. Clay

Dr. Watson, a petrochemical engineer from the University of Arkansas, stopped by the library today to talk about a two-day trip opportunity. However, if you are in 4th or 6th periods, then you mostly acquired your information through Mr. Clay, the career counselor. He also discussed ACT score and how important it is to the college application process. You need a 19 to get into the University of Arkansas unconditionally, and 18 to get in with remediation classes.

Remember, I do after school ACT prep from 3:40pm - 5:40pm in Mrs. Cain's room - 120. We cover all the subjects: Mrs. Cain does English and reading, I do math and science (aka how to read graphs properly). Come and join us - practice makes perfect!

Shout out to Chakara Adkins, Monica Horton, Chandler Baker, and Mark Robinson for coming to ACT prep today after school!

Thursday and Friday: Recreating the Mayan Sundial

Just as the Mayans used trigonometry to tell time with sundials, we can also use ourselves to determine what (relative) time of day it is. On Thursday, we went outside and measured our shadows while facing the sun. Students went home and got their own measurements at a different time of the day, using the same procedure as in class. Then, on Friday, we explored the relationship between the two triangles created at the two different times of day.

Our bodies represented the height of the triangle, our shadows were the base, and the hypotenuse was the distance between our head and the tip of the shadow. Students should have determined that shadows are longer at dawn/evening and the shortest right around noon. This is because the angle of elevation (that is, the angle between the Earth and the sun's position) is small during dawn/evening. Our shadow represents the adjacent side. The smaller the angle becomes, the larger cos(our angle) becomes, and thus, the longer the adjacent side (our shadow) becomes.

Shout out to Kierra Robinson and Joe Tripp for making over 100% on their trig projects - they were just THAT good!

Quadrantal Angles and Missing Angles

Quadrantal Angles end on the quadrants - they are angles that are multiples of 90 degrees. If one of the sides of the angle passes through a point on the axis, we cannot draw a triangle to calculate the six basic trig functions. Thus, we have to remember in this case that:

y = the opposite side
x = the adjacent side
the distance between the origin and the point: the hypotenuse

We also calculated missing angles within a triangle. This can be done by using the 2nd - inverse functions on the calculator above the three trig functions. We can fill in all missing sides and angles of a triangle.

The second review sheet was given out today, and you know how to do everything on it! Due date is September 12th, early due date is September 11th for 10 bonus points. Friday is the last day to retake the last quiz.