Three Functions, but same idea.
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Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.
Before getting stuck into the functions, it helps to lớn give a name to lớn each side of a right triangle:
- "Opposite" is opposite to lớn the angle θ
- "Adjacent" is adjacent (next to) to lớn the angle θ
- "Hypotenuse" is the long one
Adjacent is always next to lớn the angle
And Opposite is opposite the angle
Sine, Cosine and Tangent
Sine, Cosine and Tangent (often shortened to lớn 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 lớn one decimal place):
Size Does Not Matter
The triangle can be large or small and the ratio of sides stays the same.
Only the angle changes the ratio.
Try dragging point "A" to lớn change the angle and point "B" to lớn change the size:
Good calculators have sin, cos and tan on them, to lớn make it easy for you. Just put in the angle and press the button.
But you still need to lớn remember what they mean!
In picture form:
How to lớn remember? Think "Sohcahtoa"!
It works lượt thích this:
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
You can read more about sohcahtoa ... please remember it, it may help in an exam !
Angles From 0° to lớn 360°
Move the mouse around to lớn see how different angles (in radians or degrees) affect sine, cosine and tangent.
In this animation the hypotenuse is 1, making the Unit Circle.
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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.
"Why didn't sin and tan
go to lớn the party?"
"... just cos!"
Example: 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 √3:
Now we know the lengths, we can calculate the functions:
|sin(30°) = 1 / 2 = 0.5|
|cos(30°) = 1.732 / 2 = 0.866...|
|tan(30°) = 1 / 1.732 = 0.577...|
(get your calculator out and kiểm tra them!)
Example: what are the sine, cosine and tangent of 45° ?
The classic 45° triangle has two sides of 1 and a hypotenuse of √2:
|sin(45°) = 1 / 1.414 = 0.707...|
|cos(45°) = 1 / 1.414 = 0.707...|
|tan(45°) = 1 / 1 = 1|
Why are these functions important?
- Because they let us work out angles when we know sides
- And they let us work out sides when we know angles
Example: Use the sine function to lớn find "d"
- The cable makes a 39° angle with the seabed
- The cable has a 30 meter length.
And we want to lớn know "d" (the distance down).
Start with:sin 39° = opposite/hypotenuse
sin 39° = d/30
Swap Sides:d/30 = sin 39°
Use a calculator to lớn find sin 39°: d/30 = 0.6293...
Multiply both sides by 30:d = 0.6293… x 30
d = 18.88 to lớn 2 decimal places.
The depth "d" is 18.88 m
Try this paper-based exercise where you can calculate the sine function for all angles from 0° to lớn 360°, and then graph the result. It will help you to lớn understand these relatively simple functions.
You can also see Graphs of Sine, Cosine and Tangent.
And play with a spring that makes a sine wave.
Less Common Functions
To complete the picture, there are 3 other functions where we divide one side by another, but they are not ví commonly used.
They are equal to lớn 1 divided by cos, 1 divided by sin, and 1 divided by tan:
|sec(θ) = HypotenuseAdjacent||(=1/cos)|
|csc(θ) = HypotenuseOpposite||(=1/sin)|
|cot(θ) = AdjacentOpposite||(=1/tan)|
1494, 1495, 724, 725, 1492, 1493, 726, 727, 2362, 2363
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