Three Functions, but same idea.
Graphing Sine And Cosine FunctionsRight Triangle
Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.
The main difference between the two is that cosine wave leads the sine wave by an amount of 90 degrees. A sine wave depicts a reoccurring change or motion. It is known as sine wave as it has the similar shape as the sine function, when it is plotted on a graph. The graph shows the repetition of one wave segment in a repeated manner. So, you need to graph a sine, cosine, or tangent function. Sine, cosine, and tangent â and their reciprocals, cosecant, secant, and cotangent â are periodic functions, which means that their graphs contain a basic shape that repeats over and over indefinitely to the left and the right. SWBAT sketch a graph of a tangent function and compare the features graph to those of sine and cosine. Big Idea By exploring the graph of the tangent function, students get a preview of some important ideas that they may see in future math classes. Sine and cosine graphs y = sin x and y = cos x look pretty similar; in fact the main difference is that the sine graph starts at (0,0) and the cosine at (0,1).
Before getting stuck into the functions, it helps to give a name to each side of a right triangle:
Adjacent is always next to the angle
And Opposite is opposite the angle
Sine, Cosine and Tangent
Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle:
For a given angle θ each ratio stays the same
no matter how big or small the triangle is
To calculate them:
Divide the length of one side by another side Example: What is the sine of 35°?
Using this triangle (lengths are only to one decimal place):
Good calculators have sin, cos and tan on them, to make it easy for you. Just put in the angle and press the button.
But you still need to remember what they mean!
In picture form:
Practice Here:Sohcahtoa
How to remember? Think 'Sohcahtoa'!
It works like this:
You can read more about sohcahtoa .. please remember it, it may help in an exam !
Try It!
Have a try! Move the mouse around to see how different angles (in radians or degrees) affect sine, cosine and tangent.
In this animation the hypotenuse is 1, making the Unit Circle.
Notice that the adjacent side and opposite side can be positive or negative, which makes the sine, cosine and tangent change between positive and negative values also.
ExamplesExample: what are the sine, cosine and tangent of 30° ?
The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of â3:
Paint tool sai cracked version.
Now we know the lengths, we can calculate the functions:
(get your calculator out and check them!)
Example: what are the sine, cosine and tangent of 45° ?
The classic 45° triangle has two sides of 1 and a hypotenuse of â2:
Why?
Why are these functions important?
Example: Use the sine function to find 'd'
We know:
And we want to know 'd' (the distance down).
sin 39° = d/30
Use a calculator to find sin 39°: d/30 = 0.6293..
d = 18.88 to 2 decimal places.
The depth 'd' is 18.88 m Exercise
Try this paper-based exercise where you can calculate the sine function for all angles from 0° to 360°, and then graph the result. It will help you to understand these relatively simple functions.
You can also see Graphs of Sine, Cosine and Tangent.
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Less Common Functions
To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used.
They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan:
What is cosine of 3?
Undefined!!!! Can't answer it! All sine and cosine values are between -1 and 1 !!!
If two integers have the same sign what is the sine of their sum?Graphing Sine Cosine Tangent Comparison Chart
Sine(A+ B) = Sine(A)*Cosine(B) + Cosine(A)*Sine(B).
Why it is named sinusoidal wave or sine wave in ac?
It's called a sine wave because the waveform can be reproduced as a graph of the sine or cosine functions sin(x) or cos (x).
What are the Differentiate the sine wave and cosine wave?
The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.
How do you find the cosine and the sine?
Sine= Opposite/ Hypotenuse Cosine= Adjacent/ Hypotenuse
Sine Cosine And Tangent CalculatorHow does the tangent function relate to sine and cosine?Graphing Sine Cosine Tangent Comparison Formula
Tangent = sine/cosine provided that cosine is non-zero. When cosine is 0, then tangent is undefined.
Why are sine and cosine functions used to describe periodic?What is the derivation of the wave function?
A simple wave function can be expressed as a trigonometric function of either sine or cosine. lamba = A sine(a+bt) or lamba = A cosine(a+bt) where lamba = the y value of the wave A= magnitude of the wave a= phase angle b= frequency. the derivative of sine is cosine and the derivative of cosine is -sine so the derivative of a sine wave function would be y'=Ab cosine(a+bt) '''''''''' cosine wave function would beâ¦
Graphing Sine Cosine Tangent Comparison CalculatorHow do you find sine cosine and tangent of 210 degrees?
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