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Course: Trigonometry>Unit 2

Lesson 5: Trigonometric values on the unit circle

Sine & cosine identities: symmetry

Sal finds several trigonometric identities for sine and cosine by considering horizontal and vertical symmetries of the unit circle. Created by Sal Khan.

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• Do I have to memorize all of these formulas? Is it important enough to know?
• No, you can get through a lot of math without memorizing, but it just takes a lot longer to do the problems. Sometimes it is just plain easier to memorize a couple of formulas than to try to dig back to the basics and reconstruct the formulas.

In the case of the symmetry relationships, it is a great time-saver to know these. There are ways of reconstructing the information if you forget. One way is to memorize the signs for the different trig functions in the four quadrants. The way I remind myself of these formulas is to think of a point in the first quadrant (both x and y will be positive, so all sine and cosine values will be positive, as will tangent).

Then I think of a point in the second quadrant (x will be negative, since all the values for x will be less than zero, and y will be positive. As a result, sine will be positive, but cosine will be negative, and all tangent values will be negative.) In the third quadrant, all x and y values will be negative, so all sine and cosine values will be negative. Tangent will be positive because a negative divided by a negative is positive.) The final quadrant is the fourth quadrant, and there, all x values are positive, but all y values are negative, so sine will be negative, cosine will be positive and tangent values will be negative.

So, you CAN recreate the information by logic. In the meantime, others can use the symmetries and be done with the problem and maybe with the next problem as well. Also, there are some ways that the questions can be asked that make it difficult to use this method, and if you are not very conversant with unit circle and translating points to sine and cosine, then you may have some tough slogging ahead. Knowing this set of symmetries becomes handy in Physics and many other applications.
• What is a radian? I think I have some vague description of a ratio and something with PI but I don't honestly remember.
• A radian is the angle you get with its vertex at the center of a circle, and the two line segments that contain the angle each connect the vertex to an endpoint of an arc on the circle, and the arc length is equal to the radius of the circle. The arc is opposite the angle. Converting to degrees, a radian is approximately 57.3 degrees; pi radians equals 180 degrees.
• so, this is basic trig right? still amaze me with its complications! i still don't get it :( any websites that helps for all basic trig ?? and college math ... thanks!
(1 vote)
• Honestly, you don't need any other websites. I have been looking at the Khan Academy for Trig, and it does a great job on explaining. Maybe just reviewing it over and over will help.
• I must say that was a lot to take in for one video, usually there's the odd "There's this reason" or "There's that" , not much made alot of sense to me with without examples or references .
• I thought it was mostly review, but just pulled together as a different way of looking at the symmetries around a unit circle. So, what exactly was confusing? There are a number of more basic videos on trigonometry that may help, but I am not sure where to recommend that you start. Then after you are familiar with the basics, it would be easier to see how the concepts fit together. The key was showing how to name the reflected angles, and then how to construct the cosine and sine statements for each angle, and finally how to relate the pairs. And it all starts with the unit circle, so if you are hazy on that, it would be a great place to start your review.

For example, let's say that we are looking at an angle of π/3 on the unit circle. The value of `sin (π/3) is ½√3` while `cos (π/3) has a value of ½`
The value of
`sin (-π/3) is -½√3` while `cos (-π/3) has a value of ½`
Already we can see that `cos theta = cos -theta` with this example.
And look at that: `sin -theta = -sin theta` just like Sal said.
If we go through all the other reflected angles, with this specific example of an angle, we will get the same relationships that Sal just walked us through.
One other example is π - theta
This looks mysterious until you realize that in our situation, theta equals π/3
and π - π/3 gives us an angle of ⅔π (In degrees, that is 120 degrees)
cos ⅔π is going to be a negative value because it is in the second quadrant, and from working with unit circles, we should remember that it is evaluated as cos π/3 except for the negative. So, cos (π - π/3) = - cos π/3 and `cos π/3 = - cos (π - π/3)`
Basically, if you have these symmetries, you have a multitude of sine and cosine values as long as you know what sine of theta is and cosine of theta is.

It may help you to continue around the circle with common angles like π/6 and π/4 (not to mention the rest of the π/3 gang). That may help you to see how the symmetrical relationships are great time savers.
• why sin(-theta)=-sin(theta)
• Sin(theta) produces a positive value on the y axis.

-Sin(theta) or, the negative of Sin(theta) takes that value, and multiplies it by -1, giving us a negative value on the y axis.

That same negative value on the y axis, can be produced with Sin(-theta)
• What is SOH CAH TOA?
• (S)ine
(O)pposite
(H)ypotenuse
(C)osine
(A)djacent
(H)ypotenuse
(T)angent
(O)pposite

The first letter in each of the letter triplets names the ratio the second letter of each triplet is the numerator, the third letter of each triplet is the denominator.
• at , wouldnt cos of theta be positive because the x is positive? and the sin be negative because the y is negative? why are they both negative?
• It's only the angle, or theta, that's negative. If you were to simplify the negative angles, then cos(-θ) = cos(θ) and sin(-θ) = -sin(θ).
• what is the diffrence between -data and +data
• - Theta is when you rotate clockwise to the right.
+Theta is when you rotate counterclockwise to the left.
• I know this might be a dumb question but is Theta?
• theta is just the variable used when referring to an angle a lot of the time
• Why, why, why was this video placed AFTER solving sinusoidal equations?

You literally CANNOT solve them without these identities, but they don't teach you said identities until after you spend an hour failing to solve those equations and moving on. Staff members, please consider repositioning this video in the curriculum.
• A few times when I took KA's courses I noticed weird orderings like that. You should try making a request at their help center at Zendesk for your concern to get noticed.

Happy learning,
- Convenient Colleague
(1 vote)