Question

geometry help? will medal and fan.

Explain the difference between using the trigonometric ratios (sin, cos, tan) to solve for a missing angle in a right triangle versus using the reciprocal ratios (sec, csc, cot). You must use complete sentences and any evidence needed (such as an example) to prove your point of view.

Answer 1

Sine Function: sin = Opposite / Hypotenuse

Cosine Function: cos = Adjacent / Hypotenuse

Tangent Function: tan = Opposite / Adjacent

Cosecant Function: csc = Hypotenuse / Opposite

Secant Function: sec = Hypotenuse / Adjacent

Cotangent Function: cot = Adjacent / Opposite

Answer 2

Where are you stuck? It looks like you nailed all six trig ratios perfectly.

Though I would say instead of "sin = opposite/hypotenuse" I would write "sin(angle) = opposite/hypotenuse"

I would do the same for each of the other trig functions as well.

Answer 3

@jim_thompson5910 I'm stuck on explaining how they differentiate for solving a missing angle :(

Answer 4

So normally you're probably used to knowing the angle and using the angle, plus another side, to find the missing side

eg: you know the angle is 30 degrees and and the hypotenuse is 4. Using sin(angle) = opp/hyp, you can figure out that the opposite side is 2

Answer 5

if you don't know the angle, then you would leave it as x or theta

let's say the angle is x

Answer 6

if the hypotenuse is 4 and the opposite side is 2, then

sin(angle) = opp/hyp

sin(x) = 2/4

sin(x) = 1/2

now is the question: how to isolate x? Well we just apply the inverse operation of sine. Called `inverse sine` or `arcsine`

Answer 7

Thanks! so how would I explain this in literary form? :P

Answer 8

you mean in terms of words instead of symbols?

Answer 9

yes

Answer 10

I would go through each step I wrote above and try to translate. For instance.

`The angle is unknown, so we'll call it x for now. The sine of the angle, aka sin(x), is equal to the ratio of the opposite over the hypotenuse. In other words, sin(x) = opp/hyp. Because the opposite side is 2 units and the hypotenuse is 4 units, the sine ratio is equal to 2/4 = 1/2. To isolate x, we apply arcsine to both sides and that will lead to x being 30 degrees. `

That's one way to do it. Of course there are infinitely many other ways. So that should give you an idea

Answer 11

ok, I'll do that. thank you!

Answer 12

no problem

Answer 13

wait what do I do with arcsine lol

Answer 14

you would use a calculator at this point. Your calculator would have an arcsine button, or an inverse sine button

The button usually looks like \(\Large \sin^{-1}\) as shown in the attached image

Answer 15

In the case of that ti83 plus image I showed, you would hit the yellow `2nd` key first, then hit the `sin` button then type in 0.5 or 1/2 followed by a closing parenthesis. Then hit enter

Answer 16

Make sure your calculator is in degree mode

Answer 17

I did, thanks!

Answer 18

@math_genius12345 @jim_thompson5910

Note that teh 'reciprocal function' is NOT the same as the 'Inverse fuction'

sec, cot and csc are all ratios of side lengths exactly like sin cos and tan

sin^-1 etc is an ANGLE

TBH I cannot see what th point of this question is - there is no real difference in using any of the functions if you know all the sides of a triangle

Answer 19

I guess you could argue that sin cos and tan appear on most calculators and as functions in most programming languages, spreadsheets etc.

But there is no inherent mathematical reason why sin is better than cosec.

Sin and cos are limited to range +/- 1 but tan goes to infinity - so there is no real benefit there.

This question is difficult to justify having an 'opinion'