Basic trigonometry help, please

[Lutherf]

I've got a niece coming to me for high school math help and she's on her introduction to trig. She has the formulas to define sin, cos and tan but keeps going to her calculator and plugging in whatever spits out. I know that back in the day I had to do these calculations by hand so I would understand WHY the sin31° was .515 but I'll be damned if I can remember how to do the calculation.

 

[radioman]

I've got a niece coming to me for high school math help and she's on her introduction to trig. She has the formulas to define sin, cos and tan but keeps going to her calculator and plugging in whatever spits out. I know that back in the day I had to do these calculations by hand so I would understand WHY the sin31° was .515 but I'll be damned if I can remember how to do the calculation.

Hmmmm....as I recall, in a right triangle, the sine is the ratio of the length of the side opposite the angle divided by the length of the hypotenuse of the same right triangle.
Fun fact....knowledge of the sine function was useful to me in building house foundations on hillsides.

 

[mike2810]

quick method

 

[Chomsky]

I've got a niece coming to me for high school math help and she's on her introduction to trig. She has the formulas to define sin, cos and tan but keeps going to her calculator and plugging in whatever spits out. I know that back in the day I had to do these calculations by hand so I would understand WHY the sin31° was .515 but I'll be damned if I can remember how to do the calculation.

You probably forgot that we used the trig tables in the back of the algebra book! Remember, now? Long tables, maybe a half dozen columns per page, continuing in rows down the page - for many pages?

Anyway, memorize

'SCOT'

and

'Old Houses Are Houses Of Angels'!

--

S = Sine = Opposite over Hypothenuse = O/H = 'Old Houses'
C = Cosine = Adjacent over Hypothenuses = A/H = 'Are Houses'
O
T = Tangent = Opposite over Adjacent = O/A = 'Of Angels'

--

Remember, applying the trig function (Sin, CoS, Tan) to the angle (in degrees), returns the ratio of the sides. Conversely, the inverse (or 'arc') function applied to the ratio, returns the angle in degrees!

--

There ya' go! You & the kid memorize that little 'Scot' ditty, along with being cognizant the trig functions work fully reciprocally, and you're most of the way there!

Good luck!

 

[Chomsky]

Hmmmm....as I recall, in a right triangle, the sine is the ratio of the length of the side opposite the angle divided by the length of the hypotenuse of the same right triangle.
Fun fact....knowledge of the sine function was useful to me in building house foundations on hillsides.

Awesome!

My first application of trig as a practical exercise, was to calculate putting shelves in my bedroom in a 90* corner. It was complicated by my using multiple boards to build the shelves out, for several feet, using multiple 8" boards. So, this meant I had all sort of angles between the room walls & the boards themselves, to practice on.

I decided to not do any trial fits, but just move ahead when I found the first board fit, and all the numbers cross-checked every-which way. And, I'll be damned if it all didn't work perfectly with the remaining boards. Quite honestly, that little exercise gave me the desire to pursue math more seriously as I entered college.

- -

And get this:

My best buddy was in a union carpenter apprenticeship program, at the time. They know how to do this stuff using practical 'hands-on' methods with physical tools, but of course do not use or know the trig functions. He was blown away when I showed him how I did the calculations. We had a blast together, with this. That's when I knew I wanted to be an engineer, not a tradesman. I knew what my buddy was doing as he showed me his techniques, but he had no idea what he was doing - besides knowing how to do it as taught. Whereas I could see the whole picture, and how everything fit together.

Anyway, that's my story & I'm stickin' to it!

 

[Lutherf]

quick method

Right. That much I've got. The problem, however, is that the questions generally don't have dimensions for all sides. For example, If you want to know how tall a telephone pole is by measuring the shadow and having the angle from you to the top then you only have an angle and one side of a triangle. In that case you'd need to know that the sine of whatever angle you're standing at is some specific number. So now i need to calculate the sine of, for example, 65°. At this point the kid is just plugging sin65° into her calculator and solving the equation. That's great but how the heck do you figure out that the sine of 65° is .906 WITHOUT a calculator?

 

[radioman]

Anyway, that's my story & I'm stickin' to it!

And an interesting story it is.
Funny....I was always a whiz at math, but was more interested in the "nuts and bolts" application of math.
Just like your friend, I became a tradesman.
I think my love of numbers waned when "integrals" and "derivatives" came along. LOL

 

[Phys251]

Right. That much I've got. The problem, however, is that the questions generally don't have dimensions for all sides. For example, If you want to know how tall a telephone pole is by measuring the shadow and having the angle from you to the top then you only have an angle and one side of a triangle. In that case you'd need to know that the sine of whatever angle you're standing at is some specific number. So now i need to calculate the sine of, for example, 65°. At this point the kid is just plugging sin65° into her calculator and solving the equation. That's great but how the heck do you figure out that the sine of 65° is .906 WITHOUT a calculator?

Trig ratios are just that, ratios. Solving sin(65°) without a calculator would be one messy trail of trig identities.

 

[nota bene]

I think I've posted about this before, but I'd like to give thanks yet again to Ivan, the oceanography doctoral student (with an emphasis on plankton--I remember with such gratitude!) who taught my trig class and gave me a "gentlewoman's C" so that I could finally finish my first degree. I was a 5th-year senior because I lacked one math class, and I did not deserve to pass this one, but I memorized the formulas and correctly identified which one was needed to answer a question.

Bless you, Ivan, and I paid it forward.

 

[Chomsky]

That's great but how the heck do you figure out that the sine of 65° is .906 WITHOUT a calculator?

Luther, did you read my post upthread?

Before calculators you used the tables in the back of the textbook!

Read my description in post #4 to refresh your memory . . .

 

[mike2810]

As Chomsky said. If available use sin tables.

Downloadable Trig Table PDF - Sine, Cosine, Tangent

This trig table PDF contains values for sine, cosine and tangent for angles between 0 and 90º. All values are rounded to four decimal places.

sciencenotes.org

A little more math.

How does calculator find the sine of an angle?

www.homeschoolmath.net

This is the best I can do for you. Too old . This is making my head ache.

 

[Chomsky]

And an interesting story it is.
Funny....I was always a whiz at math, but was more interested in the "nuts and bolts" application of math.
Just like your friend, I became a tradesman.
I think my love of numbers waned when "integrals" and "derivatives" came along. LOL

I was nuts & bolts too, but sucked at math because I never worked at it! That changed in college, with the help of a great TA. He convinced me to work hard, under his tutelage, and that's what it finally took.

I will say this about the trades: It's amazing how they lend themselves to moonlighting & working for yourself!

When my kids were young, I pressed them to get their degrees. I encouraged the one that seemed trade bound, too. But I strongly encouraged him to get his degree first, then enter the trades - but only enter if if he intends to learn on their dime, and eventually work for himself & own the place.

I think that's the way to do really well in the trades, by working for yourself & starting a business. Me & my childhood buddies came up blue collar working class, and many went into trades. Of those, the union guys seem to do alright. But the guys that really kicked ass, were those went out on their own. Some are amazing stories, and they are really cleaning house! The ones that hit it big, all either never worked for anyone from the start, or switched to working for themselves very early-on. When you're young and live with your folks, you can afford to work on your own for whatever comes your way. And with these guys, a real lot came their way! I mean a REAL lot!

 

[EdwinWillers]

Couldn't have survived all my applied math courses without this:

Not from the above, but for reference:

 

[Ug make hammer]

I've got a niece coming to me for high school math help and she's on her introduction to trig. She has the formulas to define sin, cos and tan but keeps going to her calculator and plugging in whatever spits out. I know that back in the day I had to do these calculations by hand so I would understand WHY the sin31° was .515 but I'll be damned if I can remember how to do the calculation.

I can't really help you since back in my school days we used trig tables. It was a "must be in the syllabus for students with no calculator" despite my class all having calculators. The teacher would go get a second-hand calculator from the store-room if a kid really couldn't wrangle one from their parents.

No wait. There MUST be a way to calculate trigs by hand. They're all a sum of an infinite series, so ask first "to how many decimal places?" and then add up enough of the fractions to get that close.

Don't forget the 3/4/5 right triangle, it's a crowd pleaser. Also, there are dozens of valid proofs of Pythagoras' Theorem. You could give you neice some hints and let her "invent" one of them for herself.