Танген

Practice Here:

Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.

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Divide the length of one side by another side

In picture form:

Move the mouse around to see how different angles (in radians or degrees) affect sine, cosine and tangent.

Size Does Not Matter

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They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan:

Если у Вас имеются какие-либо предложения или замечания, мы будем рады узнать о них.

To calculate them:

How to remember? Think «Sohcahtoa»!

In this animation the hypotenuse is 1, making the Unit Circle.

Sohcahtoa

Possible causes are:

To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used.

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Using this triangle (lengths are only to one decimal place):

But you still need to remember what they mean!

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.

Angles From 0° to 360°

Before getting stuck into the functions, it helps to give a name to each side of a right triangle:

Try this paper-based exercise where you can calculate the sine functionfor all angles from 0° to 360°, and then graph the result. It will help you to understand these relativelysimple functions.

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The triangle can be large or small and the ratio of sides stays the same.

It works like this:

The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of√ 3 :

Example: what are the sine, cosine and tangent of 45° ?

Good calculators have sin, cos and tan on them, to make it easy for you. Just put in the angle and press the button.

Now we know the lengths, we can calculate the functions:

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Why are these functions important?

The classic 45° triangle has two sides of 1 and a hypotenuse of √ 2 :

Try dragging point «A» to change the angle and point «B» to change the size:

(get your calculator out and check them!)

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Why?

For a given angle θ each ratio stays the same
no matter how big or small the triangle is

Only the angle changes the ratio.

Possible causes are:

And we want to know «d» (the distance down).

You can read more about sohcahtoa. Remember it, as it may help in an exam !

And Adjacent is always next to the angle

The depth «d» is 18.88 m

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Example: what are the sine, cosine and tangent of 30° ?

Opposite is always opposite the angle

Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle:

© 2014 – 2024 Ing. Adam Kašpárek, Jihlava, Czech Republic, IN: 02394260

Example: Use the sine function to find «d»

Examples

Источники:

https://uchim.org/matematika/tablica-tangensov&rut=85b984db3e067bed9516b1149e59194c91360783f152b48b9ac5b74b0953d7ae
https://www.calculat.org/ru/%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8/%D1%82%D0%B0%D0%BD%D0%B3%D0%B5%D0%BD%D1%81/&rut=d8634e89e5d30fe3d0a717054e8316b619ee62e4a8fb29a18b63d2c39dd4e6d9
https://en.wikipedia.org/wiki/Tangent&rut=07003c6b3e726f4658b5fccf018723556fc76411a03c83a42aa4efd8a8189081
https://en.wikipedia.org/wiki/Trigonometric_functions&rut=2673a0151c151f0841abd4c549bc1c742296b2198919bddd87008db335b9b998
https://www.mathsisfun.com/sine-cosine-tangent.html&rut=2a4a703ee7abb98b16779197d100e4bd765cb028284f559e4d54e7943e4edcdb
https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8&rut=2de151dc7eed62afa22538ffb6c947a86447e479236695227610bcc8586f762e
https://mathworld.wolfram.com/Tangent.html&rut=5a33538a290228e75760dc1207a504588a71b1ead18721b520e99f25043dcfd1
https://www.calculat.org/en/trigonometric-functions/tangent/&rut=052967480f35c5ac0f4a0f2c415003c2b24cd81de87101234097ac858e122941
https://www.youtube.com/watch?v=edknVqkxQW4&rut=97a8a613530521456840d5de2c90c01443d43544cae9eaebe49d9a3eed566e21