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## Algebra II (2018 edition)

### Course: Algebra II (2018 edition)>Unit 9

Lesson 2: The unit circle definition of sine, cosine, and tangent

# Unit circle

Learn how to use the unit circle to define sine, cosine, and tangent for all real numbers. Created by Sal Khan.

## Want to join the conversation?

• Do these ratios hold good only for unit circle? What if we were to take a circles of different radii?
• The ratio works for any circle. The advantage of the unit circle is that the ratio is trivial since the hypotenuse is always one, so it vanishes when you make ratios using the sine or cosine.
• What is the terminal side of an angle?
• straight line that has been rotated around a point on another line to form an angle measured in a clockwise or counterclockwise direction
• So how does tangent relate to unit circles? And what is its graph?
• I think the unit circle is a great way to show the tangent. While you are there you can also show the secant, cotangent and cosecant. I do not understand why Sal does not cover this.
Using the unit circle diagram, draw a line “tangent” to the unit circle where the hypotenuse contacts the unit circle. This line is at right angles to the hypotenuse at the unit circle and touches the unit circle only at that point (the tangent point). Extend this tangent line to the x-axis. The distance of this line segment from its tangent point on the unit circle to the x-axis is the tangent (TAN). If you extend the tangent line to the y-axis, the distance of the line segment from the tangent point to the y-axis is the cotangent (COT). To determine the sign (+ or -) of the tangent and cotangent, multiply the length of the tangent by the signs of the x and y axis intercepts of that “tangent” line you drew. For example, If the line intersects the negative side of the x-axis and the positive side of the y-axis, you would multiply the length of the tangent line by (-1) for the x-axis and (+1) for the y-axis. You will find that the TAN and COT are positive in the first and third quadrants and negative in the second and fourth quadrants.
As a bonus, the distance from the origin (point (0,0)) to where that tangent line intercepts the x-axis is the secant (SEC). The sign of that value equals the direction, positive or negative, along the x-axis you need to travel from the origin to that x-axis intercept. The distance from the origin to where that tangent line intercepts the y-axis is the cosecant (CSC). The sign of that value equals the direction positive or negative along the y-axis you need to travel from the origin to that y-axis intercept.
Some people can visualize what happens to the tangent as the angle increases in value. When the angle is close to zero the tangent line is near vertical and the distance from the tangent point to the x-axis is very short. As the angle nears 90 degrees the tangent line becomes nearly horizontal and the distance from the tangent point to the x-axis becomes remarkably long. You can, with a little practice, “see” what happens to the tangent, cotangent, secant and cosecant values as the angle changes.
The second bonus – the right triangle within the unit circle formed by the cosine leg, sine leg, and angle leg (value of 1) is similar to a second triangle formed by the angle leg (value of 1), the tangent leg, and the secant leg. In this second triangle the tangent leg is similar to the sin leg the angle leg is similar to the cosine leg and the secant leg (the hypotenuse of this triangle) is similar to the angle leg of the first triangle. When you compare the sine leg over the cosine leg of the first triangle with the similar sides of the other triangle, you will find that is equal to the tangent leg over the angle leg. Therefore, SIN/COS = TAN/1 . You can also see that 1/COS = SEC/1 and 1^2 + TAN^2 = SEC^2. A bunch of those almost impossible to remember identities become easier to remember when the TAN and SEC become legs of a triangle and not just some ratio of other functions. The angle line, COT line, and CSC line also forms a similar triangle.
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When you graph the tangent function place the angle value on the x-axis and the value of the tangent on the y-axis. At the angle of 0 degrees the value of the tangent is 0. At 45 degrees the value is 1 and as the angle nears 90 degrees the tangent gets astronomically large. You are left with something that looks a little like the right half of an upright parabola. At negative 45 degrees the tangent is -1 and as the angle nears negative 90 degrees the tangent becomes an astronomically large negative value. This portion looks a little like the left half of an upside down parabola. This pattern repeats itself every 180 degrees. Do yourself a favor and plot it out manually at least once using points at every 10 degrees for 360 degrees. It may not be fun, but it will help lock it in your mind. Give yourself plenty of room on the y-axis as the tangent value rises quickly as it nears 90 degrees and jumps to large negative numbers just on the other side of 90 degrees.
• I hate to ask this, but why are we concerned about the height of b? What is a real life situation in which this is useful? Graphing sine waves?
• Say you are standing at the end of a building's shadow and you want to know the height of the building. you only know the length (40ft) of its shadow and the angle (say 35 degrees) from you to its roof. you could use the tangent trig function (tan35 degrees = b/40ft)
40ft * tan35 = b
28ft = b
Now you can use the Pythagorean theorem to find the hypotenuse if you need it.
• In the concept of trigononmetric functions, a point on the unit circle is defined as (cos0,sin0)[note - 0 is theta i.e angle from positive x-axis] as a substitute for (x,y). This is true only for first quadrant. how can anyone extend it to the other quadrants? i need a clear explanation... I think trigonometric functions has no reality( it is just an assumption trying to provide definition for periodic functions mathematically) in it unlike trigonometric ratios which defines relation of angle(between 0and 90) and the two sides of right triangle( it has reality as when one side is kept constant, the ratio of other two sides varies with the corresponding angle).... i think mathematics is concerned study of reality and not assumptions.... how can you say sin 135*, cos135*...(trigonometric ratio of obtuse angle) because trigonometric ratios are defined only between 0* and 90* beyond which there is no right triangle... i hope my doubt is understood..... if there is any real mathematician I need proper explanation for trigonometric function extending beyond acute angle.
• [cos(θ)]^2+[sin(θ)]^2=1 where θ has the same definition of 0 above.
This is similar to the equation x^2+y^2=1, which is the graph of a circle with a radius of 1 centered around the origin. This is how the unit circle is graphed, which you seem to understand well.
Based on this definition, people have found the THEORETICAL value of trigonometric ratios for obtuse, straight, and reflex angles. This value of the trigonometric ratios for these angles no longer represent a ratio, but rather a value that fits a pattern for the actual ratios. I hope this helped!
Proof of [cos(θ)]^2+[sin(θ)]^2=1:
• This seems extremely complex to be the very first lesson for the Trigonometry unit. He keeps using terms that have never been defined prior to this, if you're progressing linearly through the math lessons, and doesn't take the time to even briefly define the terms. No question, just feedback.
• The problem with Algebra II is that it assumes that you have already taken Geometry which is where all the introduction of trig functions already occurred. So Algebra II is assuming that you use prior knowledge from Geometry and expand on it into other areas which also prepares you for Pre-Calculus and/or Calculus. So if you need to brush up on trig functions, use the search box and look it up or go to the Geometry class and find trig functions.
• At , shouldn't the point on the circle be (x,y) and not (a,b)? [Since horizontal goes across 'x' units and vertical goes up 'y' units--- A full explanation will be greatly appreciated]
• It would be x and y, but he uses the letters a and b in the example because a and b are the letters we use in the Pythagorean Theorem `a²+b² = c²` and they're the letters we commonly use for the sides of triangles in general. It doesn't matter which letters you use so long as the equation of the circle is still in the form `a²+b² = 1`.
• You should at least explain what sin and cosine are. This is intro to trigonometry