Review the unit circle definition of the trigonometric functions.
What is the unit circle definition of the trigonometric functions?
The unit circle definition allows us to extend the domain of sine and cosine to all real numbers. The process for determining the sine/cosine of any angle is as follows:
- Starting from , move along the unit circle in the counterclockwise direction until the angle that is formed between your position, the origin, and the positive -axis is equal to .
- is equal to the -coordinate of your point, and is equal to the -coordinate.
A unit circle on an x y coordinate plane where the center of the unit circle is at the origin and the circumference of the circle touches (one, zero), (zero, one), (negative one, zero), and (zero, negative one). Point (cosine of theta, sine of theta) is located near one thirty o'clock on the circle. A line segment extends from (zero, zero) to the point. An angle from the arc starting at (one, zero) and opening counter clockwise to the point has an angle measurement of theta degrees. A dashed line extends vertically from the point to the positive x axis. The length from the pont to the x axis is sine of theta. The length from where the dashed line touched the x axis to (zero, zero) is cosine of theta.
The other trigonometric functions can be evaluated using their relation with sine and cosine.
Want to learn more about the unit circle definition? Check out this video.
Appendix: All trig ratios in the unit circle
Use the movable point to see how the lengths of the ratios change according to the angle.
Check your understanding
A unit circle on an x y coordinate plane where the center of the unit circle is at the origin and the circumference of the circle touches (one, zero), (zero, one), (negative one, zero), and (zero, negative one). Point (sixty four hundredths, seventy seven hundredths) is located near one thirty o'clock on the circle. A line segment extends from (zero, zero) to the point. An angle from the arc starting at (one, zero) and opening counter clockwise to the point has an angle measurement of fifty degrees.
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- In the appendix, can someone explain to me WHY cotangent, tangent, secant, and cosecant are represented in that way?(4 votes)
- tangent=sin(θ)/cos(θ)(tangent is the path from point(x,y) to the secant)
cotangent=1/tan(θ)(cot α 1/tan)(cotangent is the path from the point (x,y) to the cosecant)
secant=1/cos(θ)(sec α 1/cos)
cosecant=1/sin(θ)(csc α 1/sin)(18 votes)
- In the Appendix (moving point example above) I have difficulty relating the tangent (in blue) to a line drawn from the center of the circle through the green dot. The tangent would only match if the x-intercept of the sine (red line) was equal on both sides of the red line.(6 votes)
- Do you have a full picture of a unit circle... please help with this :)(5 votes)
- So if sine, cosine, and tangent are SOH CAH TOA, could we say that secant, cosecant, and cotangent are SHO CHA TA0?(2 votes)
- Almost. Secant is the reciprocal function of cosine, and cosecant that of sine, so it would actually read CHO SHA CAO. Probably not best to memorize them like this, though, because you already have one trig mnemonic and they mightget jumbled up. Plus, cosecant and cotangent start with the same letter. Hope this helps.(7 votes)
- Why is it that you don't use the actual angles when finding tangent(3 votes)
- It's just easier to think of it as sin/cos. After all, the unit circle only defines cosine and sine.(4 votes)
- I understand how to apply the trig ratios in their purest form (absolute and a standard right angle triangle).
I also understand the theory of using the unit circle to extend the domain of sin(theta) and cosine(theta) to all real values.
With that said, how is a calculator able to take this basic principle and compute the logic that cos(89) = 0.017 and cos(91) = -0.017 (a reflection across the y-axis). What's going on under the hood?(4 votes)
- Hi kimberm1,
I would assume that they created an algorithm that checks the angle just like a human would.
Aka first calculate the absolute value of cos(theta), and then check to see what the value of theta is.
Here is some PSUEDO CODE that might help you understand.
IF (theta >= 0 AND theta < 90) THEN
ANSWER = cos(theta) // leave unchanged
IF (theta >= 90 AND theta < 180) THEN
ANSWER = cos(theta) * -1
And so on. What the computer would essentially do is follow an algorithm that checks what quadrant the point will be in based upon the angle. We can easily calculate that as seen above.
Knowing the quadrant, we know know what sign the y and x values are.
Hope this helps,
- Convenient Colleague(1 vote)
- Why and how are trigonometric ratios related to circles?(1 vote)
- That's the subject of this video: https://www.khanacademy.org/math/algebra2/trig-functions/unit-circle-definition-of-trig-functions-alg2/v/unit-circle-definition-of-trig-functions-1, particularly around2:25.
The cool thing about a unit circle (a circle with radius one), is that if you draw a right triangle where the hypotenuse of the triangle connects a point on the unit circle with the point (0, 0), the length of the hypotenuse is always going to be one. Because that's the radius of the circle! And the length of the other two sides are just going to be the x and y coordinates of the point on the unit circle.
Take a little time drawing right triangles were the hypotenuse connects the middle of the unit circle to a point on the circle and see if you can convince yourself that this is the case.
So you have a right triangle where one side length is the x-coordinate and one side length is the y-coordinate, and the hypotenuse is equal to 1. This means you can figure out the sine and cosine of the angle that the hypotenuse and the x-axis make:
The cosine is the length of the adjacent side (which is the x-coordinate) divided by the length of the hypotenuse (which is 1). So the cosine is just the x-coordinate!
Similarly, the sine is the length of the opposite side (which is the length of the y-coordinate) divided by the length of the hypotenuse (which is 1). So the sine is the y-coordinate.
The trigonometric ratio defining the tangent (opposite over adjacent) may be less intuitive from the unit circle, but I hope you can see from the unit circle why the tangent of an angle is equal to the sine of the angle over the cosine.
Hope this helps.(5 votes)
- In the "Appendix: All trig ratios in the unit circle" above, why is the Tangent at a negative slope?(2 votes)
- Normally the unit circle is pictured with the hypotenuse going from the origin to the point where the ray intersect the circle (meaning that the right-angle corner in on the X axis) whereas the one pictured here runs through point of intersection to the X and Y axis outside of the unit circle. Pictured this way, it allows for one to see all the trig function versus just the "Big 3" (sin, cos, and tan).
That being said. The way this hypotenuse is oriented gives it a negative slope.(2 votes)
- what's an easy way to remember that sin = y-coordinate, cos = x-coordinate, and tan = y-coordinate/x-coordinate??(2 votes)
- You could use basic trigonometry if you forget using 90 degree triangle the related angle have r as 1 and defining the point (x, y) where the ray(2 votes)
- Is there exist more intuitive vision on this?
I'm confused with the extention.
I got that in first quadrant it's all work good, and sin=y, cos=x and we can definitely work with angles there, but how do we can relate 225 degrees angle to 45? For example, before, we deliberated about relationships of adjecents and opposites sides and it exactly what sin or other trigonometric function is, ratio, reletionships, but now, if we will scale our vision up to angle 225 degrees, how we can proof that cos 225 = cos 45 without only unit circle, becouse if we use our protractor, for example, this angles looks absolutely different and, of course, angle 225 degrees hasn't any reletionships between sides, 'cose it isn't right triangle anymore.(1 vote)