What are the values of sin, cos and tan?

Question

At school I've only been told to type them into my calculator whenever I need to use them. They must have some sort of value, though, otherwise we wouldn't be able to get to the answer. So what are those values?

Answer 1

They don't have values, they are operations, which means each one tells you to do something different (and shame on your teacher for not explaining this to the class BEFORE having you do them on your calculator).

You always have to have right triangles when dealing with these three trig functions (that's what sine, cosine and tangent are, trig functions). They are each ratios, which means a comparison of two numbers. Sine means to take the length of the opposite side (the side opposite the angle you're measuring) and put that number in a fraction as the numerator, then in the denominator put the length of the hypotenuse of the triangle. So for short, sine means "opposite over hypotenuse." Similarly, cosine means "adjacent over hypotenuse" and tangent means "opposite over adjacent." The adjacent side is the side forming the angle that is NOT the hypotenuse (this is really difficult to explain without being able to draw you a picture). My students called it SOHCAHTOA for short to remember them all.

I'll try and walk you through an example as best I can...Draw an isoceles right triangle (which means the triangle has one right angle and two 45 degree angles). Since it's isoceles, the two sides that are NOT the hypotenuse must be the same length. So let's say those two sides each have a length of 1. If you do the pythagorean theorem (a squared plus b squared equals c squared) you'll get that the length of the hypotenuse is equal to the square root of 2, or approximately 1.414. Did you draw the pictures and label all the parts?

Now pick one of those 45 degree angles and circle it. Let's calculate the sine of 45 degrees, WITHOUT using a calculator. Using that angle you chose, find the length of the opposite side (the side of the triangle that isn't touching the angle at all). It's 1 right? So put that in a fraction as the numerator, the number on top. Sine means opposite over hypotenuse, so in the denominator we need the length of the hypotenuse, which is the square root of 2 or 1.414. So your equation now says....

sin 45=1/1.414

On your calculator, punch in 1 divided by 1.414 and you'll get the number 0.70721....

Make sure your calculator is in degree mode, if it's not you'll get the wrong answer, it's a very common mistake!

Now punch in sin 45 and hit enter. What number do you get? 0.70710....

The reason the two numbers are different is because we rounded the length of the hypotenuse. If we'd actually used the square root function on our calculators and typed in the square root of 2 instead of the decimal, they would have been exactly the same number.

And it doesn't matter what the lengths of the sides of the triangles are, as long as they're right triangles. IF YOU DON'T HAVE RIGHT TRIANGLES THOUGH, YOU CAN'T USE THESE FUNCTIONS. So be careful. But that's all the sine, cosine and tangent represent. They are the ratios of the side lengths of right triangles.

Here's a final example. The tangent of 51 degrees (tan 51) equals 1.23489..... That means that in a right triangle with a 51 degree angle, if you take the length of the side opposite the 51 degree angle and divide it by the length of the side adjacent to that angle (remember, tangent means "opposite over adjacent"), then you'd get 1.23489....

I hope that helped. Good luck! I'm sorry I couldn't draw you a picture.

Answer 2

Sin (Sine) and Cos (Cosine) are best represented as waves e.g. a sine wave or cosine wave. A cosine wave is the same shape as a sine wave but it is half a wave out.

If you draw the following
1
|
0-----------90-----------180---------270----------360
|
-1

A sine wave follows the path of
0,0 curv to 1,90 curv to 0,180 curv to -1,270 curv to 0,360
And a cosine wave
1,0 curv to 0,90 curv to -1,180 curv to 0,270 curv to 1,360
The top and bottom halfs of the curvs are semicircles.

If the horizontal line represents degrees then at 90 degrees sine is 1 and cos is 0, at 270 degrees sine is -1 and cos is 0.

You can use sin and cos to draw a circle on graph paper by calculating the X,Y positions for each angle and plotting it.
At angle 0, x = sin(angle) and y = cos(angle). x = 0, y = 1
At angle 90, x = sin(angle) and y = cos(angle). x = 1, y = 0
At angle 180, x = sin(angle) and y = cos(angle). x = 0, y = -1
At angle 270, x = sin(angle) and y = cos(angle). x = -1, y = 0
If you complete this using all the angles from 0 to 359 then you will get your circle. By multiplying the coords by a radius and adding an offset you can change the size and position of the circle.

Tan is the tangent of a line. This is represented by drawing a curv and then drawing a straight line so that it touches the curv at some point. If the curv were a circle and you drew a line from the center so that it bisectes the line at 90 degrees and then you drew another line from the center so that it crosses the line at some other point, you can work out the distance between the 2 points on the line if you know the angle at the center.

Sine, cos and tan can then be used in trignometry to calculate unknown lengths and angles. It is used extensively in 3D computer graphics to calculate movement, positions and rotations.

Using trignometry you can take any right angled triangle with any 2 lengths or angles or even 1 angle and 1 length (not including the 90 degree angle) and using the following equations :-
Tan x = Opposite / Adjacent
Cos x = Adjacent / Hypotonuse
Sin x = Opposite / Hypotonuse
Work out the rest.

Answer 3

First of all, to help you, your question is incomplete. Imagine a right angle (90 degrees) triangle. Label the vertices (corners) ABC. Make the 90 degree corner A and this is angle A. The other vertices are B and C and the associated angles are angles B and C. Now let's look at either angle B or C. It doesn't matter which so we will choose angle B. The side opposite angle B is side AC and is called the opposite. The side next to angle B is side AB and is called the adjacent. Now, for any given value of angle B (in degrees) we find that the ratio of the Length of side AC to side BA is constant. It doesn't matter how long the sides actually are. For a given size angle the ratio of AC (opposite) to BA (adjacent) is always the same. The ratio AC:BA is called the tangent of the angle. Somebody has worked out what that ratio is for all sizes of the angle. So when you ask your calculator to give you the tan of 45 degrees what it displays is the value of that ratio. Likewise, the ratio of BA (adjacent) to BC (hypotenuse) is called the cosine or cos. The ratio of AC (opposite) to BC (hypotenuse) is called the sine or sin. Again someone has worked out the value all of these ratios for different size angles. Before the advent of calculators we used to use books of tables of sin, cos and tan values. There are others ways to explain this involving circles but I hope this helps.

Answer 4

they have no actual values. in a unit circle (a circle with 0,0 at the center, and a 1 unit radius), the sine is the y value, the cosine is the x value, and the tangent is y/x. in some calculators, you can type sin/cos/tan, etc, of some value, but it does not have a finite value.

Answer 5

Each angle has it"s own value.The easiest of all are:
Sin 30 ° is 0,5 cos 30 ° is √3/2, sin 60 º is 0,5, cos 60 º is √3/2; tg 30 º is √3/3
and tg 60 º is √3; Sin 45 ºand cos 45 ºis√2/2; tg 45 ºis 1, and so on.

Answer 6

U just need to input the sin/cos/tan of the angle you are looking for.
I think "those values" are the angles.
your calculator must be either in the degree or radian mode.

Answer 7

It's sine, cosine, and tangent.

They don't have values, they're more like an operation.

For example, the sine of 90 degrees, is 1.

Answer 8

sin has a very low value as its bad, cos i think is slightly higher as its a tasty lettuce and tan is worth the most as it warms you up in winter.

sorry no idea lol

Answer 9

if you graph it you will get the best representation of what is the y value for a certain x value...for example if you graph sin(x) you will see that sin(0) =0...or if you graph cos(x) you will see that cos(0) =1...same goes for tan(x)

Answer 10

In a right angled triangle, meaning θ is acute ie 0° < θ < 90°

sinθ = (length of opposite side)/(length of hypotenuse)
cosθ = (length of adjacent side)/(length of hypotenuse)
tanθ = (length of opposite side)/(length of adjacent side)

If the angle becomes any angle at all then (cosθ, sinθ) are the x and y coordinates of a radius rotating and starting on the positive x axis on a circle of radius 1. A positive angle is formed with an anticlockwise rotation and a negative angle with a clockwise rotation.
In this case tanθ = slope of the radius.

Note: In fact tanθ equals the length of the segment on the tangent from the point on the end of the radius (and hence on the circle) to the x-axis. If you measure in a forward direction from the radius it is positive and if you measure in a backward direction from the radius it is negative. Hence the name "tangent".