Question

can someone explain to me what this means? give a careful definition of the trigonometric functions defined on "proper angles"(angles which can be part of a right triangle)

Answer 1

I think I might be able to. Lemme just double check the definition of a proper angle to be on the safe side.

Answer 2

whoops wrong question lol

Answer 3

Hmm... Don't see a definition for that.

But...

Sine of an angle, let's call it "theta", will be the number you get by dividing the side that is on the opposite side of the triangle by the hypotenuse (the longest side, it's opposite of the 90 degree angle). So, if your triangle is a right triangle and has these sides: 3,4,5 and you pick the angle that is opposite of the side that has 3, your sin of that angle would be 3/5.

The cosine of the angle is the adjacent side (the touching side that is not the hypotenuse) divided by the hypotenuse. So for our triangle it is 4/5.

Finally, the tangent will be the only remaining combo: opposite and adjacent. The tangent of an angle is the answer that you get when you divide the opposite by the adjacent, or 3/4 for our triangle.

Hope this is what you wanted?

Answer 4

hey liz u got it?

Answer 5

kind of. so sin(t)=y, cos(t)=x, tan(t)=y/x where x cannot=0. So sin of an angle equals opposite over hypotenuse=y/1=y and cosine of an angle= adjacent over hypotenuse=x/1=x and tangent of an angle=opposite over hypotenuse=y/x where x cannot =0. Is this the best way to explain it using these words

Answer 6

sine is opposite by hypotenuse
cos is adjacent by hypotenuse
tan is opposite by adjacent

Answer 7

get it?

Answer 8

I do a little. I have in my notes. Any ratio we make from the side lengths of a right triangle is invariant under similarity, so depends on the angles, not the actual triangle. what does this mean

Answer 9

it means the ratio is same for a given angle anywhere

Answer 10

That works, dizliz. The reason you can use x or y is because you can draw a triangle on a coordinate plane (or a graph as it's more commonly called). But it's a good idea to remember opposite/hypotenus/adjacent as that's how triangles are more commonly encountered. A trick to remember them:

cos without the o is cs. cs sounds like x. So it's x/hypotenuse.
sin sounds like sYn. So sine is Y/hypotenuse.

:)

Answer 11

does that mean that if we enlarge the triangle that the ratio is the same

Answer 12

yes surely

Answer 13

That's right!

Answer 14

so which angles can be part of a right triangle

Answer 15

any angle can be

Answer 16

Thing of it like this:
No matter how big you make a square, they will always have the same ratio to the sides (as long as the angles remain the same). They'll always be 1:1:1:1.

Likewise, if you have an equilateral triangle, the sides will always be 1:1:1.

If you have a 30:60:90 triangle, the sides will always be 1:2:3 (divide each amount by 30).

Answer 17

Any angle below 90 can be.

Answer 18

30, 60, 90 triangle doesnt have sides in 1:2:3

Answer 19

@Him: If the angle is 90 or more, then it's no longer a real/proper triangle.

Answer 20

Oh right. It's 1:2:sqrt(3) for my example if I recall correctly.

Answer 21

dont confuse her now

Answer 22

liz come to chat

Answer 23

Did you have any other questions, Liz?

Answer 24

I have a ton of questions. I have an exam tom at 9 and I have about four more qeustions on this take home quiz

Answer 25

dont worry well help

Answer 26

That is helpful, so I can draw a triangle to prove this

Answer 27

yeah

Answer 28

So if I draw a triangle with sides 1,2,3 and enlarge it by a factor of 2 and prove it works. Will that help answer this question

Answer 29

Yeah, that'd work. Only difference would be that each side would be twice as long.

Answer 30

The angles would still remain the same.

Answer 31

yeah
in ur first case u write down all the three ratios.....and then in the second case all ur sides are multiplied by 2..so the ratios wil remain the same..get it?

Answer 32

do u understand liz?

Answer 33

It's kind of like how no matter how big or small you make a square, they'll always be the same, as long as all the sides are enlarged by the same ratio.

But if you decide to change the ratio for even one side, you'll either get a trapezoid or a rectangle or just a random quadrilateral.

Answer 34

k

Answer 35

did u unerstand the half angle formula?

Answer 36

No I need that explained to me. Give me one sec to finish this question

Answer 37

u there?

Answer 38

liz?

Answer 39

Yes. Now how would i answer this question. Suppose f is any function defined on the set of all real numbers with the properties: f is one to one; f has image equal to the set of all positive real numbers; f satisfies the addition law f(x+y)=f(x) f(y) for all real numbers x and y

Answer 40

whts the question?

Answer 41

? dont take 12 weeks to reply please

Answer 42

Sorry, show that f has an inverse function g, with domain the set of positive real numbers, and show that g satisfies g(xy)= g(x)+g(y) for all positive real numbers x and y

Answer 43

my computer is acting funny again

Answer 44

there are two ways to do this/....do u know derivatives?

Answer 45

no. we used exponentials and logarithms

Answer 46

tell me where you started off...cz the long part is obtaining the function from these conditions...did u go to exponents directly?

Answer 47

I think so

Answer 48

so see

the function

f(x) = a^x

is a function which satisfies ur conditions..check em one by one..and tell me doesnt it?

Answer 49

funny comp agn?

Answer 50

no just thinking

Answer 51

I want to show f using exponents and g using its inverse log

Answer 52

im coming to that..so are we decided that f satisfies ur conditions?

Answer 53

Im not sure

Answer 54

f(x) = a^x

f(x)f(y) = a^(x) a^(y) = a(x+y)
ok?

Answer 55

k

Answer 56

so now do u know how to find an inverse of a function?

Answer 57

yes

Answer 58

so
y = f(x) = a^x
now find its inverse

Answer 59

Not in this form. I know how to write log for 2^3=8

Answer 60

see
\[y = a^x , \ now \ take \ \log \ \to \ the \ base \ a\ on \ both \ sides\]

you will get

\[\log_{a}y = \log_{a} (a^x) => \log_{a}y = xlog_{a}a = x \]

Answer 61

now interchange x and y

so

\[y = f(x) = \log_{a}x\]

Answer 62

understand>

Answer 63

??

Answer 64

A little better

Answer 65

dont worry now..tell me....next question....or any doubts?

Answer 66

I feel like I can't make a connection with it

Answer 67

I dont know what it is.

Answer 68

functions and stuff?

Answer 69

come to chat..its faster

Answer 70

Could we try using something with a base 2 to explain from the beginning

Answer 71

as you wish madam

Answer 72

let

\[y = f(x) = 2^x\]

ok?

Answer 73

come to chat

Answer 74

Its not working

Answer 75

it isnt

Answer 76

so is the function ok?

Answer 77

no my message isnt going through on chat

Answer 78

neither is mine

Answer 79

i need f and g to undo each other

Answer 80

they will..if 2^x is f
whts g? tell me?

Answer 81

??

Answer 82

for instance I have 2^2=4 which is log2(x)=2. Is that correct

Answer 83

yes..so the inverse fn is log2(x)..isnt it?

Answer 84

yes

Answer 85

so now see if f and g cancel out

Answer 86

How do i do that

Answer 87

see if g is the inverse of x then

f(g(x)) = x...do u understand?>

Answer 88

What numbers do I put in place of f, g,x

Answer 89

f(x) is 2^x ....g(x) is log2(x)

so f(g(x)) = 2^(log2 (x)) isnt it?

Answer 90

which is equal to x

Answer 91

almost there thinking about it

Answer 92

ah, because 2 and log2 cancel each other out. But why is that true

Answer 93

its a rule of logarithm

Answer 94

how can i show that g(xy)=g(x) +g(y)

Answer 95

bcoz log(xy) = log x + log y

Answer 96

ok?

Answer 97

no, how does that work. is it because the inverse exponentials with addition is multiplication

Answer 98

yes