# Hard time with trigonomic functions.

• Llama77
In summary: There is a simple geometric interpretation of the trig functions as well. If you are familiar with the unit circle, well these trig functions are derived from it. For instance, if you take a point (x,y) on the unit circle, and draw the line from the origin to (x,y), you have created a right triangle. Then, by definition, sin theta = y, and cos theta = x. This is why the points on the unit circle are sometimes given as (cos theta, sin theta). But there is more! The unit circle is a very nice way to remember the trig functions.
Llama77
I am currently a student in a Engineering based calculus class and well things are about average in terms of my performance, Just there is one thing that I just don't get and for that matter have never gotten. That is trig functions, I just don't understand them. I've tried memorizing and reading old textbooks, i just can't learn them. I need a way to learn these things before they kill me. Any suggestions

Llama77 said:
I am currently a student in a Engineering based calculus class and well things are about average in terms of my performance, Just there is one thing that I just don't get and for that matter have never gotten. That is trig functions, I just don't understand them. I've tried memorizing and reading old textbooks, i just can't learn them. I need a way to learn these things before they kill me. Any suggestions

Well you can think of them in a lot of different ways. I'm sure some really great suggestions will come about from this thread, so you will have to suffice with some mediocre ones for the time being (from moi).

Trig functions have really interesting properties. For example, remember this:

"SOHCAHTOA"

Where:
Sin(theta) = O/H
Cos(theta) = A/H
Tan(theta) = O/A

So these functions take the value of the angle for a ratio of a triangle. That's pretty neat.

Or how about the fact that if you paramterize a curve as, x = cos(theta) and y = sin(theta) you can feed x(theta), y(theta) a value from 0 to 2pi and draw a a circle.

Or better yet you can get into the very interesting property of representing "any" (as far as my knowledge is concerned) function as a representation of sin and cos functions via a Fourier series. That means that yes you can turn that straight line into a bunch of sin and cos functions.

With cos and sin, you can stop using time as the domain and instead use the frequency domain which makes things very nice in some situations. Phasors? Have yoiu heard of those... very nice.

Anyways, all I can say is just play around with them. The more practice you get, the better you will be. What don't you understand about them? Ask away!

I could neither understand nor memorise most of the trig functions/formulae initially, but surprisingly, I got a good grasp after working with a lot of problems in calculus involving trig. functions.

a basic problem is to measure arclength on a circle. this equivalent to measuring angles, since the angle cut out by the arc of circle is proprotional to the length of the arc.

the basic trig function "sin" is just the name given to one of these arc length functions, or rather its inverse.

i.e. think of the unit circle, and an arc beginning at (1,0) AND RUNNING UP COUNTERCLOCKWISE and ending at a point (x,y). Think of the arc length of that arc as a function of y.

that arclength function is called arcsin, ND ITS INVERSE IS SIN.

i.e. the arclength of the qarc ending at the point (x,y) is arcsin(y) = t, and if you know the arclength t of this arc, then sin(t) = y is the ehight of the endpoint.

all thi makes good sense at least for points in the first quadrant. for other points, the arcsin is not single valued, but the sin is, so we use sina s a primary function.

then cos is just sqrt(1-sin^2), and tan is sin/cos, and sec = 1/cos, and that's about all you usually need.another very useful way to view these functiions, that is a klittle more sophisticated, is as the real and imaginary parts of the complex exponential function. i.e. e^(it) = cos(t) + isin(t).

this is actually the best way to understand the complicated addition formuals for sin and cos. since e^i(s+t) = e^is e^it, by multiplying out we get:

cos(s+t) + isin(s+t) = [cos(s)cos(t)-sin(s)sin(t)] + i [cos(s)sin(t)+sin(s)cos(t)],

and setting real and imag parts equal, gives the laws:

cos(s+t) = [cos(s)cos(t)-sin(s)sin(t)] and
sin(s+t) = [cos(s)sin(t)+sin(s)cos(t)].

these are hard to remember otherwise.also cos and sin are important as basic solutions to the differential equation

f'' + f = 0. try e^it in there also and see why these functions are related.

i myself missed trig in high school and did not know any when i got to college (and no calculus), where fortunately for me they were presented ab initio using power series.

mathwonk said:
the basic trig function "sin" is just the name given to one of these arc length functions, or rather its inverse.

I love when high level math individuals go through the basics! It's very interesting as well as helpful to see how they think about certain topics. Thanks! mathwonk for your post, I especially enjoyed what I quoted above from you. It's interesting to think of them as arc length functions.

The sine of an angle is a function which assigns to each angle the ratio of the opposite side to the hypotenuse as if the angle was in a right triangle.

The cosine of an angle is is a function which assigns to each angle the ratio of the adjacent side to the hypotenuse as if the angle was in a right triangle.

The tangent of an angle is a function which assigns to each angle the ratio of the opposite side to the adjacent side as if the angle was in a right triangle. Thus it is also equivalent to the ratio of sine to cosine.

$$\frac{O/H}{A/H} = \frac{O}{A}$$

However, these only work for acute angles. For obtuse angles, sin and cosine are defined in respect to the complements of these angles.

Now, many of the properties of the trigonometric functions arise from these simple definitions.

For instance: the area of any triangle as (1/2) AB sin theta

is a direct consequence of the definition of area as (1/2)(Base times height). This is because, if B is the base of the triangle, A times sin theta is the height.

Also, the pythagorean identity is a direct consequence of the pythagorean theorem (hence the name).

EDIT: Of course, to give definite values to the trig functions, one needs power series (except in the simple cases such as 0, 30, 45, 60, and 90 degrees).

Last edited:
im a little confused on what you guys mean by a ratio. so is Sin a ration of Oppositeypotenuse and if so what does that mean.sorry for the delay I have been real busy.

the length of the side opposite side from the angle divided by the length of the side of the longest side (the hypotenuse) is the sine of the angle in a right angle. Try that and verify it is correct.

Sine and cosine are easy.

Step 1) Take a circle with of radius 1 centered at the origin.

Step 2) Start at the coordinate (1, 0) -- (this would be 3 o'clock on a clock).

Step 3) Walk around the circle in a counter-clockwise direction.

Step 4) After you have traveled for a distance of x units, you are at the new coordinate (cos x, sin x). (The parameter x is in radians, not degrees).

And that's one way to define cosine and sine.

Some examples of using this definition.

* If you stay in place, you have traveled 0. You started at (1, 0) and you didn't move anywhere, so cos 0 = 1 and sin 0 = 0.

* If you walk all the way around the circle, you have traveled the entire perimeter, and again, you end where you started. The perimeter has a length of 2 * pi. So cos(2 * pi) = 1, sin(2 * pi) = 0.

* If you walk half way around the circle, you are going from (1, 0) to (-1, 0). Half the perimeter is pi, so cos pi = -1 and sin pi = 0.

* If you walk a quarter of the way around the circle, you are going from (1, 0) to (0, 1). A quarter of the perimeter is pi / 2, so cos(pi / 2) = 0 and sin(pi / 2) = 1.

I hope that helps a bit.

## 1. What are trigonometric functions?

Trigonometric functions are mathematical functions that relate angles in a right triangle to the lengths of its sides. The most commonly used trigonometric functions are sine, cosine, and tangent.

## 2. Why do people have a hard time with trigonometric functions?

Trigonometric functions can be challenging for some people because they involve complex mathematical concepts, such as angles and ratios, and require a solid understanding of geometry and algebra.

## 3. How can I improve my understanding of trigonometric functions?

To improve your understanding of trigonometric functions, it is important to practice solving problems and familiarize yourself with the properties and applications of each function. You can also seek help from a tutor or teacher if needed.

## 4. Are there any real-world applications of trigonometric functions?

Yes, trigonometric functions have many practical applications in fields such as engineering, physics, astronomy, and navigation. They are used to calculate distances, angles, and heights in various real-world scenarios.

## 5. What are some tips for mastering trigonometric functions?

Some tips for mastering trigonometric functions include practicing regularly, understanding the fundamental concepts, memorizing important formulas and identities, and breaking down complex problems into smaller, more manageable parts.

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