Question 1

What are these new ratios used for?

Accepted Answer

Eventually, in calculus, you will need sec(x), csc(x), and cot(x) for the derivative (rate of change) of some of the trigonometric functions. In particular, the first derivative of tan(x) is (sec(x) )^2

Question 2

What is cosh, sinh, and tanh? I saw these functions on the calculator.

Accepted Answer

Those are hyperbolic functions:
sinh x = ½(e^x - e^(-x))
cosh x = ½(e^x + e^(-x))
tanh x = sinh x / cosh x = (e^x - e^(-x)) / (e^x + e^(-x))

Question 3

Wouldn't it make more sense for "secant" to be sin, and "cosecant" cos? There is no good reason for it to be the other way around than to absolutely troll us

Accepted Answer

I know, right?

Question 4

Why do you need these functions if you already have sine, cosine, and tangent?

Accepted Answer

Strictly speaking, we don't these days.  Historically speaking, finding trig values and reciprocals were much much harder than pressing two buttons on a scientific calculator.  So people wanted to have separate tables for looking up 1/sin x and so on.  In fact, those weren't the only "extra" trig tables people had back then.  Check out this fun article!
http://blogs.scientificamerican.com/roots-of-unity/10-secret-trig-functions-your-math-teachers-never-taught-you/
So, why do we still hold on to secant, cosecant, and cotangent when we dropped stuff like havercosine and excosecant?  A good reason is that they make the trig formulas in calculus a little easier to remember and use, and also because the geometric meaning of the secant can be valuable at times.  But other than that, they totally take a back seat to the three principal trig functions.

Question 5

I don't like using Soh Cah Toa. First, the 'h' is silent in two of them, so that's a first layer of decoding it in your head. Then, they're single letters that you have to plug into their original term, which is another layer of decoding, and then you have to go through and put it all together. To me it's a messy mnemonic.
I prefer:
"Opp Hy" (sounds like 'up high')
"Add Hy" ('add height')
"Opp Add" (like an 'op ed' article)

It's easy to start the rhythm with:
"Sine Opp Hy" (sounds like 'Sign up high') easy
then you just know that cosine comes after sine, and tangent is last because we all know that Op Ed articles can go off on tangents.

I dunno - it works a billion times better for me than Soh Cah Toa *shrugs*

Accepted Answer

I think your alternate mnemonic works great! The whole point is to help you remember facts (in this case, the relationships between the trig functions and the sides of a right triangle).

The best mnemonics are the ones you make up that are meaningful to you, because they will be easier to remember and work better.

Question 6

How would you find the sin, cosine, or tangent of 90 degrees?

Accepted Answer

Well, sin of 90 degrees means that you are trying to find the opposite over hypotenuse of a triangle with a measure of 90 degrees. But the opposite side to a 90 degree angle IS the hypotenuse. We usually tackle these angles when we have moved to the unit circle because they don't fit the Soh Cah Toa rule. Sin of 90 degrees is one. Cos of 90 degrees is 0, because if the angle has rotated through 90 degrees, there is nothing left of the adjacent side. Tangent is the sine divided by the cosine, so if Sine of 90 degrees is one and Cosine of 90 degrees is zero, you have 1/0 which is undefined

In fact, you get an error message if you ask a calculator to give you tangent of 90.

In fact, you get an error message if you ask a calculator to give you tangent of 90.

Question 7

In the sections for secant and cosecant, the article said that secant is the reciprocal of cosine and that cosecant is the reciprocal of sine. Isn't it supposed to be the other way around?

Accepted Answer

An easy way to remember is that each group only has one "co"

Question 8

So what is the exact difference between cosecant, secant, cotangent and cos-1, sin-1, tan-1?

Accepted Answer

cosecant, secant and tangent are the reciprocals of sine, cosine and tangent.
sin-1, cos-1 & tan-1 are the inverse, NOT the reciprocal. That means sin-1 or inverse sine is the angle θ for which sinθ is a particular value. For example, sin30 = 1/2. sin-1 (1/2) = 30. 
For more explanation, check this out.
https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles/trig-solve-for-an-angle/a/inverse-trig-functions-intro

Question 9

Are there any other ratios we will learn in the future?

Accepted Answer

Well, the textbook answer is that there are only 6 trig ratios, which we have already covered. However, if you really want to devel into the topic, the historical answer would be that there are at least 12 ratios, which include the ones we've learned and some new ones which are versine, haversine, coversine, hacoversine, exsecant, and excosecant. They can be expressed as the following:
 `versine(θ) = 2 sin2(θ/2) = 1 – cos(θ)
haversine(θ)= sin2(θ/2)
coversine(θ)= 1 – sin(θ)
hacoversine(θ)= 1/2(1-sin(θ))
exsecant(θ)= sec(θ) – 1
excosecant(θ)= csc(θ) – 1`

	Verbal description	Mathematical relationship
cosecant	The cosecant is the reciprocal of the sine.	$\csc (A) = \frac{1}{\sin (A)}$ ‍
secant	The secant is the reciprocal of the cosine.	$\sec (A) = \frac{1}{\cos (A)}$ ‍
cotangent	The cotangent is the reciprocal of the tangent.	$\cot (A) = \frac{1}{\tan (A)}$ ‍

Trigonometry

Course: Trigonometry > Unit 1

Reciprocal trig ratios

The cosecant $(\csc)$ ‍

The secant $(\sec)$ ‍

The cotangent $(\cot)$ ‍

How do people remember this stuff?

Finding the reciprocal trigonometric ratios

Let's study an example.

Solution

Finding the cosecant

Finding the secant

Finding the cotangent

Try it yourself!

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