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### Course: Get ready for AP® Calculus > Unit 1

Lesson 13: Unit circle introduction# The trig functions & right triangle trig ratios

Sal shows how, for acute angles, the two different definitions of the trigonometric values (SOH CAH TOA and the unit circle definition) result in the same values. Created by Sal Khan.

## Want to join the conversation?

- Hard to pay attention to, all I see is letters and those don't add or divide. they are not numbers. letters everywhere and confused(42 votes)
- Think of them as numbers with no identity, they are called variables.(15 votes)

- The video doesn't really help with the practice exercises, i've seen similar complaints below that are from a year ago and the issue has not been fixed since then... i'm jumping to the next section for now but please fix the issue, so we can come back and finish this section.

I've got two exercises out of five right in a row and as soon as i think to have got it, something blows up, it's so frustrating...(17 votes)- I found that watching the videos on Symmetry and periodicity of trig functions https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry/trig-symmetry-periodicity/v/trigonometry-unit-circle-symmetry really helped with the Unit Circle Trigonometry practice.(14 votes)

- Still kinda confused on the fractions on the unit circle, hard to remember.(18 votes)
- is to illustrate the fact their either the unit circle or a plain triangle can be used to find the same trig ratios(6 votes)

- In the unit circle, can we imagine the hypotenuse in any rt angle triangle is a straight line on a cartesian plane and so it's equation is y=mx+c, and, since c = zero, m=y/x - the equation for tan(theta). So tan(theta) is the gradient of the hypotenuse?(11 votes)
- Yes, your reasoning is perfectly right. Tangent is defined by the Y ordenade of the point in which the line defined by the angle crosses the x=1 line.(3 votes)

- What does Sal mean by the Unit Circle is an extension of SOH-CAH-TOA? I know what they mean, but how is it an extension?(3 votes)
- In SOH, CAH, TOA, definition of trig functions, we defined the sin,cos etc .

as the ratio of the sides of a triangle. Also, we were only able to find the value of trig functions of angles upto 90 degrees.

But in unit circle definition, the trigonometric functions sine and cosine are defined in terms of the coordinates of points lying on the unit circle x^2 + y^2=1. And unlike the previous definition(SOH, CA...), we are able to find sin,cos.... of all possible angles.

That's how the Unit Circle definition is an extension of SOH,CA.... definition.(9 votes)

- Not much explanation of what the ratios are or how they are used. Just a lot of quick talk. The exercise questions do not follow the lesson and have that bland text-book feel to them when you use hints. Khan Academy has definitely moved away from the simple approach that it was founded on to a more public school diploma chugging factory approach...(7 votes)
- In a physical sense, how can sin be applied to anything 90 or above, or 0 or below? Same thing for Cos and Tan?(5 votes)
- trigonometric functions like sine, cosine, and tangent have broad applicability in physical scenarios that involve angles both greater than 90 degrees and less than 0 degrees. These functions help describe various aspects of oscillatory motion, periodic phenomena, slopes, and ratios, making them essential tools in many scientific and engineering disciplines. The choice of angle measurement, whether in degrees or radians, depends on the specific context and conventions used in a particular field of study.

When dealing with angles greater than 90 degrees, the sine function is used to model oscillatory phenomena that extend beyond the initial position. For example, in mechanical engineering, you might use sine functions to describe the motion of a pendulum that swings past its vertical equilibrium point.

Negative angles can represent situations where you are measuring an angle in the opposite direction. For example, if you have a rotating object that starts at a certain position and moves counterclockwise, you can use negative angles to measure its position relative to the starting point. The sine function can still be used to calculate the vertical displacement from this position.

The cosine function is used to describe phenomena that involve periodic motion or oscillations. For instance, in electrical engineering, you might use cosine functions to model alternating current (AC) voltage or current, which oscillates above and below zero.

like with sine, negative angles can be used to measure positions in the opposite direction. The cosine function can still be employed to calculate horizontal displacements or projections in these cases.

Tangent is used to describe the steepness or slope of a line, including lines that are nearly vertical. In physics, this can be applied when studying inclined planes, projectile motion, or other scenarios involving angles greater than 90 degrees.

Negative angles can be used to describe directions opposite to the initial reference direction. Tangent can be applied to calculate the slope or ratio of vertical to horizontal displacement for angles in these directions.(3 votes)

- How can I solve for a trigonometric function without using my calculator? What does mathematicians do to solve a sine or inverse sine function in middle ages?(4 votes)
- According to Wikipedia, "Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering."(4 votes)

- Still kinda confused on the fractions on the unit circle, hard to remember.(3 votes)
- Which fractions are you talking aobut, the trig fractions as shown by SOH CAH TOA, fractions created by points at various angles such as 30, 60, 90, etc. or the fractions in radians related to these degree angles? There are all sorts of fractions related to the unit circle. I also notice that this is not even your own question, it is directly copied from the second most popular post, come up with origin question because directly copying and acting like it is your own is plagiarism.(5 votes)

- What happens if there is no unit circle? Do you just have to use a calculator or is there another way?(4 votes)
- While the unit circle is a useful visual aid for understanding and calculating trigonometric functions, it's not the only way to work with trigonometry. Trigonometric functions can be defined and calculated independently of the unit circle, and you can use other methods, including calculators and tables, to determine their values.

Here are some alternative methods for calculating trigonometric functions when you don't have access to a unit circle:

Calculator: Modern scientific and graphing calculators have built-in trigonometric functions that can directly compute the sine, cosine, tangent, and other trigonometric values for any given angle. You can enter an angle in degrees or radians, and the calculator will provide the corresponding trigonometric value.

Trigonometric Identities: Trigonometric identities, such as the Pythagorean identities and sum/difference identities, can be used to express trigonometric functions in terms of other trigonometric functions. These identities can be helpful for simplifying trigonometric expressions and solving trigonometric equations.

Trigonometric Tables: Before the advent of calculators, trigonometric tables were widely used to look up values of trigonometric functions for specific angles. These tables provide precomputed values for common angles, and you can interpolate between values for angles that are not listed.

Trigonometric Formulas: Trigonometric functions can be expressed in terms of complex numbers using Euler's formula

This approach is particularly useful in advanced mathematics and engineering, as it allows you to work with complex exponentials to solve various problems.

Numerical Methods: In some cases, when dealing with non-standard angles or complex problems, numerical methods like approximation algorithms or calculus techniques may be employed to calculate trigonometric values with precision.

While the unit circle provides an intuitive geometric interpretation of trigonometric functions, it's not necessary for performing trigonometric calculations. In practice, calculators and trigonometric tables are the most common tools for obtaining trigonometric values quickly and accurately.(2 votes)

## Video transcript

Voiceover:On the right-hand side we have a bunch of expressions
that are just ratios of different information
given in these two diagrams. Then over here on the left we have the sine taken of angle MKJ, cosine of angle MKJ, and
tangent of angle MKJ. Angle MKJ is this angle right over here same thing as theta, so these two angles. These two angles have the same measure. We see that right over there. What we want to do is figure out which of these expressions are equivalent to which of these
expressions right over here. I encourage you to pause the video and try to work this through on your own. Assuming you've had a go at it, let's try to work this out. When you look at this diagram, it looks like the
intention here on the left is this evokes the unit circle definition of trig functions because
this is a unit circle right over here, and this evokes kind of
the soh cah toa definition because we're just kind of in a plain, vanilla right triangle. Just to remind ourselves, let's just remind ourselves of soh cah toa because I have a feeling
it might be useful. Sine is opposite over hypotenuse. Cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. We can refer to this and we can also remind ourselves of the
unit circle definition of trig functions that
the cosine of an angle is the X coordinate and that the sine of where this ray
intersects the unit circle, and the sine of this angle is going to be the Y coordinate. What we'll see through this video is that they actually, the
unit circle definition, is just an extension of soh cah toa. Let's look first at X over one. We have X, X is the X coordinate. That's also the length of this side right over here, relative to this angle, theta. That is the adjacent side. So X is equal to the adjacent side. What is one? Well, this is a unit circle. One is the length of the radius which for this right triangle
is also the hypotenuse. If we apply the soh cah toa definition, X over one is adjacent over hypotenuse, adjacent over hypotenuse, adjacent over hypotenuse, that's cosine. That's going to be this is
equal to cosine of theta, but theta is the same thing as angle MKJ. They have the same measure so cosine of angle MKJ is equal to cosine of theta which is equal to X over one. Now let's move over to Y over one. Well, Y is going to be
the length of this side right over here. Y is going to be, let
me do this in the blue. Y is going to be this length
relative to angle theta. That is the opposite side. That is the opposite side. Now which trig function is
opposite over hypotenuse? Opposite over hypotenuse? That's sine of theta. Sine of theta. So sine of angle MKJ is the same thing as sine of theta. We see that they have the same measure, and now we see that's the same thing as Y over one. Now for both of these I used
the soh cah toa definition, but we could have also used
the unit circle definition. X over one, that's the same thing. That's the same thing as X, and the unit circle definition says the X coordinate of where this, I guess you could say, the
terminal side of this angle, this ray right over here, intersects the unit circle. That by definition, by
the unit circle definition is the cosine of this angle. X is equal to the cosine of this angle, and the unit circle definition, the Y coordinate is equal
to the sine of this angle. We could have written
this as instead of X, Y, we could have written
this as cosine of theta, sine theta just like that,
but let's keep going. Now we have X over Y. We have adjacent over, we have adjacent over opposite. So this is equal to
adjacent over opposite. Tangent is opposite over adjacent, not adjacent over opposite. This is the reciprocal of tangent. This right over here, if we had to, this is equal to one
over tangent of theta. We later learn about
cotangent and all of that which is essentially this, but it's not one of our choices. So we can rule this one out. Then we have Y over X. Well, this is looking good. This is Y is opposite. Opposite. X is adjacent relative to angle theta. Adjacent. So this is the tangent of theta. This is equal to tangent of theta. Tangent of angle MKJ is the same thing as tangent of theta which is equal, which is equal to Y over X. Now let's look at J over K, so J over K. Now we're moving over to this triangle, J over K. Relative to this angle because this is the angle that we care about, J is the length of the adjacent side, and K is the length of the opposite side, of the opposite side. This is adjacent over opposite. This is equal to adjacent over opposite. Tangent is opposite over adjacent not adjacent over opposite. So once again this is the reciprocal of the tangent function
not one of the choices right over here so we
can rule that one out. Now K over J. Well, now this is opposite over adjacent. Opposite over adjacent. That is equal to tangent of theta. This is equal to tangent of theta, or tangent of angle MKJ. This is equal to K over J. Now we have M over J, M over J. Hypotenuse over adjacent side. This of course is equal to the hypotenuse. Hypotenuse over adjacent. Well, if it was adjacent over hypotenuse, we'd be dealing with cosine, but this is the reciprocal of that. This is actually one
over the cosine of theta not one of our choices, not one of our choices here so I'll just rule that one out over there. Then we have it's reciprocal, J over M. That's adjacent over hypotenuse. Adjacent over hypotenuse is cosine. This is equal to cosine of theta, or cosine of angle MKJ so
we could write it down. This is equivalent to J over M. Then one last one, K over M. Well, that's opposite over hypotenuse, opposite over hypotenuse. That's going to be sine of theta. This right over here is
equal to sine of theta which is the same thing
as sine of angle MKJ which is the same thing as
all of these expressions. This is equal to K over M, and we are done.