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## Class 11 Physics (India)

### Course: Class 11 Physics (India)>Unit 3

Lesson 7: Basic trigonometric identities

# Tangent identities: symmetry

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

## Want to join the conversation?

• How does the pi in tan(theta + pi) get cancelled out?
because tan(theta + pi) = tan(theta)...it doesn't make sense! •   Thinking about it as 'cancelling out' is the wrong way to look at it because, as you rightly say, you can't do that!

Instead, think that the tangent of an angle in the unit circle is the slope.

If you pick a point on the circle then the slope will be its y coordinate over its x coordinate, i.e. y/x. So at point (1, 0) at 0° then the tan = y/x = 0/1 = 0.

At 45° or pi/4, we are at an x, y of (√2/2, √2/2) and y / x for those weird numbers is 1 so tan 45° is 1.

Now you should be able to see that slope of the any ray taken from the centre point is going to be the same as another ray with which it forms a straight line, i.e. like the thick green line at .

Those two rays are always pi apart. Therefore tan(x) = tan(x+pi*n) for any value of n because the rays of all of the lines will have the same slope.

And that means we can delete the pi in tan(theta + pi) knowing that we're right.

Does that make sense?
• Wouldn't it be possible to figure out all of this without the algebra? So what is the value of doing all of this work? •   The main reason for learning to do all of these proofs is to teach you how to think logically, to go step-by-step without making unfounded assumptions, to reach a conclusion you can prove. While you are unlikely to need to prove which triangles are similar in daily life, the ability to think through problems logically is extremely important. So, geometry class is an excellent tool for teaching you logic.

• How can something plus something equal the original something? pi /= 0. 0 should be the only exception to this rule. Can someone explain? • Do I have to memorize all of these formulas? Is it important enough to know? •  Hello Chris,

I have a confession - most us us don't remember all of the formulas. Perhaps that is why the textbooks always put the trig identities on the back cover.

With that said, please do you level best to understand how these formula are derived and used. The work that you do now will be very useful as you continue through high school and move on to college.

Regards,

APD
• so is it reasonable to say that tangent will always be positive in the first and third quadrents , while tangent will always be negative in the second and fourth quadrents? • How does unit circle relate to finding out the angles of right triangles? • if you know the x value and the y value, or one of them, you can use inverse trig functions.. the radius of the unit cirlce(hypothenuse) is 1, if your x value is 1/2 you have: sin(θ)=0.5/1=0.5, then arcsin(0.5) gives you the angle(sin(30°)=1/2, arcsin(1/2)=30°), when you have both x and y values you can use inverse tangent: tan(θ)=y/x, arctan(y/x)=θ, but as the video shows, keep in mind you can have the same result from trig functions with different angles, for example: cos(θ)=cos(-θ)
• As there are so many identities is there any quick method to memorize them or do we then have to memorize each and every by heart or know how to derive them? • Well I don't know if you need to memorize all of them but (as far as I know) there is no device you can use remember them all. What I did when trying to remember these is to write them all down on one sheet of paper, not just the identities but all the really important stuff I tended to forget. When I did homework and studied I used the list like it was my bible. That's how I did it.
• Sometimes! I get confused by the wheel or the circle and the words?
so can someone just help me. • If you want any specific help, you'll need a more specific question, but my main suggestion is, if you're just having trouble understanding the unit circle, then watch the basic videos on it again. And again and again and again. And maybe search on YouTube for introduction videos on the unit circle, because maybe someone else will explain it in a way that makes more sense to you. Don't go past basic videos until you have a solid understanding.  