Sine, cosine and tangent

Sketch for a right triangle - here's how to go about it

Preliminary remark: The so-called trigonometric functions sine, cosine and tangent are nothing other than aspect ratios. In the form presented, they only apply to right-angled ones Triangles (!) and form an important basis for calculating missing pieces in the triangle. To the following explanation of this important Functions To understand, you should first prepare a tool, namely a sketch in which you enter the sizes mentioned.

• Draw a right triangle. It is best to choose it so that the hypotenuse (i.e. the longest side of the triangle) is at the bottom and the right one angle (the 90 °) are up. The two cathets are then on the left and right.
• Name the hypotenuse "c" and the left and right corners of the triangle A and B (corners have capital letters).
• The angle at A is α (alpha), the angle at B is β (beta).
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• Name the corner at the top of the triangle C, the angle there is (as already planned) 90 °.


Calculate sine beta

How can you calculate the sine of an angle, for example "Beta"? Either …

• Name the leg opposite corner A with "a", the other leg with "b".

Sine, Cosine and Tangent - a detailed explanation

• Even the mathematicians in ancient Greece noticed that all right triangles that you drew at a certain basic angle α (for example 30 °) all look similar. While these may vary in size, the shape of all of these triangles is the same.
• Ultimately, the appearance of the triangle only depends on the angle or on the relationship between the sides.
• The definitions of sine, cosine and tangent are based on this statement.
• The following applies to the sine: sin (angle) = opposite cathetus divided by the hypotenuse. "Opposite cathetus" here means the cathetus which is opposite the corresponding angle. And in this form you should also remember the definition, because the letters for the sides change yes from triangle to triangle and also in many applications you will find completely different abbreviations for the sides Select.
• For example, if the angle you are aiming for in your sketch is α, then the formula sin α = a / c results. For the angle β, however, the sine formula is sin β = b / c.
• The following applies to the cosine: cos (angle) = adjacent side divided by the hypotenuse. In this context, "adjacent cathetus" is understood to mean the cathetus lying against the angle.
• Translated into your sketch, the following applies: cos α = b / c and cos β = a / c. If you look closely, you will see that there is a connection between sine and cosine (which we will not go into here).
• The third angle function, the tangent, is required whenever the hypotenuse in the right triangle is not known. The following applies: tan (angle) = opposite side divided by the adjacent side.
• When you return to your sketch, you can implement this definition: tan α = a / b and tan β = b / a. A connection can of course also be seen here.

Sin, Cos and Tan - some examples

For the following examples and explanation you will need one calculator with the corresponding trigonometric functions. All sizes mentioned refer to the sketch.

• In a right triangle, let the hypotenuse c = 5 cm and the angle α = 35 °. With sin 35 ° = a / 5cm you can calculate the cathetus a = 2.87 cm. The leg b results from the cosine or with the Pythagorean theorem.
• In a right-angled triangle, let the two cathets a = 2.5 cm and b = 4 cm. You calculate the hypotenuse with the Pythagorean theorem. The two angles α and β result from the tangent. The following applies: tan α = 2.5 cm / 4 cm = 0.625. The inverse angle function tan^-1  (arctan or INV TAN, depending on the model) on your pocket calculator delivers the value α = 32 °. Calculate the other angle β as β = 90 ° - α = 58 °.