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### Course: Trigonometry > Unit 1

Lesson 7: 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?(77 votes)
- Sin: sine

Cos: cosine

Tan: tangent

Sec: secant

Csc: cosecant

Cot: cotangent(377 votes)

- Okay,so now we know how to calculate sin cosin and tangent.But where do we use them!!and how are these formulae derived?(17 votes)
- 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.(64 votes)

- is there any easy mnemonic device to remember which reciprocal functions and normal trig functions go together?(13 votes)
- 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!(22 votes)

- When would one want to use csc, sec and cot instead of sin, cos and tan?(19 votes)
- 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.

cot(theta)=adjacent/opposite

opposite(cot(theta))=adjacent

than it is to simplify

tan(theta)=opposite/adjacent

adjacent(tan(theta))=opposite

adjacent=opposite/(tan(theta))

I hope that makes sense!(15 votes)

- how do you do the CscA, secA amd CotA on a calculator?(9 votes)
- 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.(20 votes)

- What does reciprocal mean?(9 votes)
- 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.(13 votes)

- Isn't this the same thing as the arcsine, arcosine, and arctangent?(3 votes)
- 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.(15 votes)

- how can we say that sin square plus cos square equals to one(8 votes)
- The cosine is adjacent over hypotenuse.The sine is opposite over adjacent. So:

Cos^2(x)+sin^2(x)

(adj^2/hyp^2) + (opp^2/hyp^2)

(adj^2/opp^2)/hyp^2

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(6 votes)

- Are the csc,sec,and cot really that helpful anyway?(6 votes)
- Not really. Once you leave trig class, it's mostly sin, cos and tan from then on. That's why you don't see buttons for the other ones on calculators.(6 votes)

- hey then sal in previous video took something as tan to power of -1 but could not he take it as cot(4 votes)
- Well no. Tan^-1 is the inverse of Tan which so far I have used to find the degree of an angle using the length of the sides whereas Cot is the reciprocal which I have used to find the length of lines using the degree of an angle. Hope this helps!(4 votes)

## Video transcript

Determine the six
trigonometric ratios for angle A in the
right triangle below. So this right over here is
angle A, it's at vertex A. And to help me remember
the definitions of the trig ratios-- and these are human
constructed definitions that have ended up being very,
very useful for analyzing a whole series of
things in the world. To help me remember them, I
use the words soh cah toa. Let me write that down. Soh cah toa. Sometimes you can think
of it as one word, but it's really the three parts
that define at least three of the trig functions for you. And then we can
get the other three by looking at the first three. So soh tells us that sine of an
angle-- in this case it's sine of A-- so sine of A is equal
to the opposite, that's the O, over the hypotenuse. Well in this context, what is
the opposite side to angle A? Well, we go across the triangle,
it opens up onto side BC. It has length 12. So that is the opposite side. So, this is going
to be equal to 12. And what's the hypotenuse? Well, the hypotenuse is the
longest side of the triangle. It's opposite the
90 degree angle. And so we go opposite
the 90 degree angle, longest side is side AB. It has length 13. So this right over
here is the hypotenuse. So, the sine of A is 12/13. Now let's go to cah. Cah defines cosine for us. It tells us that cosine of
an angle-- in this case, cosine of A-- is equal
to the adjacent side to the angle over
the hypotenuse. So, what's the adjacent
side to angle A? Well, if we look
at angle A, there are two sides that
are next to it. One of them is the hypotenuse. The other one has length 5. The adjacent one is side CA. So it's 5. And what is the hypotenuse? Well, we've already
figure that out. The hypotenuse is
right over here, it's opposite the
90 degree angle. It's the longest side
of the right triangle. It has length 13. So the cosine of A is 5/13. And let me label this. This right over here
is the adjacent side. And this is all
specific to angle A. The hypotenuse would be the
same regardless of what angle you pick, but the
opposite and the adjacent is dependent on
the angle that we choose in the right triangle. Now let's go to toa. Toa defines tangent for us. It tells us that the
tangent of an angle is equal to the opposite
side over the adjacent side. So given this definition,
what is the tangent of A? Well, the opposite side,
we already figured out, has length 12. And the adjacent side,
we already figure out, has length 5. So the tangent of A, which
is opposite over adjacent, is 12/5. Now, we'll go the to the
other three trig ratios, which you could think of
as the reciprocals of these right over here. But I'll define it. So first you have cosecant. And cosecant, it's always
a little bit unintuitive why cosecant is the
reciprocal of sine of A, even though it starts
with a co like cosine. But, cosecant is the
reciprocal of the sine of A. So sine of A is opposite
over hypotenuse. Cosecant of A is
hypotenuse over opposite. And so what's the hypotenuse
over the opposite? Well, the hypotenuse is 13
and the opposite side is 12. And notice that 13/12 is
the reciprocal of 12/13. Now, secant of A
is the reciprocal. So instead of being
adjacent over hypotenuse, which we got from the
cah part of soh cah toa, it's hypotenuse over adjacent. So what is the secant of A? Well, the hypotenuse, we've
figured out multiple times already, is 13. And what is the adjacent side? It's 5. So it's 13/5, which
is, once again, the reciprocal of the
cosine of A, 5/13. Finally, let's
get the cotangent. And the cotangent is the
reciprocal of tangent of A. Instead of being
opposite over adjacent, it is adjacent over opposite. So what is the cotangent of A? Well, we've figured
out the adjacent side multiple times for
angle A. It's length 5. And the opposite side
to angle A is 12. So it's 5/12, which is,
once again, the reciprocal of the tangent of
A, which is 12/5.