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## Geometry (all content)

### Course: Geometry (all content)>Unit 13

Lesson 8: The reciprocal trigonometric ratios

# Finding reciprocal trig ratios

Sal finds all six trigonometric ratios (sine, cosine, tangent, secant, cosecant, and cotangent) of an angle in a given right triangle. Created by Sal Khan.

## Want to join the conversation?

• What is the full form of csc, sec and cot?
• Sin: sine
Cos: cosine
Tan: tangent
Sec: secant
Csc: cosecant
Cot: cotangent
• Okay,so now we know how to calculate sin cosin and tangent.But where do we use them!!and how are these formulae derived?
• 1 .Trig is used in many things in real life. Engineers, architects, astronomers, geologists, navigators, and scientists use them every day. for more, go here:
http://en.wikipedia.org/wiki/Trigonometry#Applications_of_trigonometry

2. Sine, cosine, and tangent are all mathematical operations. Your question is like asking where multiplication is derived. We came up with the 6 trigonometry functions trying to understand triangles.
• When would one want to use csc, sec and cot instead of sin, cos and tan?
• It's just for one's knowledge: also, when one has the angle and the opposite side and is trying to calculate the adjacent, it is easier to simplify the cotangent function than the tangent - this is also true for the other trig ratios trigx=a/b when you need to find b.

than it is to simplify

I hope that makes sense!
• is there any easy mnemonic device to remember which reciprocal functions and normal trig functions go together?
• Personally I just imagine that the reciprocal functions aren't allowed to start with the same letter as big three do:

sine and secant both start with s, so the inverse must be cosecant.
cosine and cosecant both start with c, so the inverse must be secant.
tangent and cotangent already start with different letters, so it works.

That's how my mind works, anyway. Maybe yours will agree!
• how do you do the CscA, secA amd CotA on a calculator?
• In the order that you gave me cscA = 1/sinA , secA = 1/cosA , cotA = 1/tanA. Thus take 1 over cos/sin/tan of the desired angle.
• What does reciprocal mean?
• 1. In reciprocal you have to take an integer (like 6) and then convert it into a fraction. In this case it would be 6/1.
2. Then switch the numerator and denominator. So your answer would be 1/6.

If the number is already fraction then just do step 2.

Hope this helps!

By the way: opposite reciprocal is the same thing, just change the positive sign to a negative or a negative sign to a positive.
• how can we say that sin square plus cos square equals to one
• The cosine is adjacent over hypotenuse.The sine is opposite over adjacent. So:
Cos^2(x)+sin^2(x)
hyp^2/hyp^2 (because of Pythagorean theorem)
1
A similar process can be done for sec^2(x)-tan^2(x)=1 and csc^2(x)-cot^2(x)=1
• Isn't this the same thing as the arcsine, arcosine, and arctangent?
• Definitely NOT. Reciprocal trig ratios are NOT the same thing as the arcsin, arccosine, and arctangent. I was screwed up by this when I was first learning trig, and it's why I really hate the notation they use to describe these concepts.

So in the video, he's talking about reciprocal ratios. Remember that when you figure out a value's reciprocal, you just "flip" it as a fraction; the numerator becomes the denominator and vice versa. And all trig functions are just ratios (fractions), so their reciprocal ratios are just the initial function "flipped." Sine is "opposite over hypotenuse (o/h)" and its reciprocal ratio is cosecant which is "hypotenuse over opposite (h/o)."

What's mixing you up is that you probably know from algebra that anything to the power of -1 has the effect of generating a reciprocal. (3/4)^-1 = 4/3. So it makes sense that what looks like sin^-1 (x) would = 1/sin(x), which is cosecant, right?

Wrong, unfortunately. What looks like sin^-1(x) is actually ARCSIN which is NOT = cosecant. Don't ask me why. It's the darn notation that's screwy - not you.

Arcsin works a lot like logarithms work, if you're familiar with those. If you ask arcsin(0.5) = ?, what you're really asking is "the sine of what angle equals .5?" In this case, the answer is pi/6 radians. (or 30 degrees). (and also 5pi/6).

sin(pi/6) = 0.5
arcsin(0.5) = pi/6 (radians)

Unfortunately, for some bizarre reason, math has chosen to represent arcsin (and the other arcfunctions) with the -1 power sign, which just confuses everything.

For the most part, you will NEVER have to deal with negative powers of trig functions. Squares, definitely - but never negative values. So if you ever see that -1 after sin, cos, or tan, just remember it represents ARCsin, ARCcos, and ARCtan and NOT the reciprocal trig functions.