As For The Sides Of The Triangle To Know The Angle | The Science



The Science

As for the sides of the triangle to know the angle

The lengths of the sides of the triangle associated with the angles at the vertices of the figure in terms of trigonometric functions -. Sine, cosine, tangent, etc. These relations are formulated in the theorems and definitions of functions in terms of the sharp corners of the triangle course of elementary geometry. Using them, we can calculate the angle for the known length of the sides of the triangle.

As for the sides of the triangle to know the angle

Instruction on how to find the angle of the sides of the triangle

Step 1:

To calculate any arbitrary angle triangle whose side lengths (a, b, c) are known, using the cosine theorem. She claims that the square of the length of any of the parties is the sum of the squares of the lengths of the other two, which is subtracted from twice the product of the lengths of the two sides of the cosine of the angle between them. Use this theorem to calculate the angle can be in any of the peaks, it is important to know only its location relative to the sides. For example, to find the angle alpha, which lies between the sides b and c, the theorem should be written as: a² = b² + c² - 2 * b * c * cos (α).

Step 2:

Give the formula of the cosine of the desired angle: cos (α) = (b² + c²-a²) / (2 * b * c). To both sides, apply the inverse function of cosine - arc cosine. It allows the value of the cosine of the angle restore degrees: arccos (cos (α)) = arccos ((b² + c²-a²) / (2 * b * c)). The left part can be simplified and the calculation formula of the angle between the sides b and c will become the final form: α = arccos ((b² + c²-a²) / 2 * b * c).

Step 3:

In the determination of the acute angles in right-angled triangle knowing the lengths of all the sides is not necessary, only two of them. If these two sides - catheti (a and b), that divide the length of which lies opposite the desired angle (α), by the length of the other. So you get the value of the tangent of the desired angle tg (α) = a / b, and applying to both sides of the inverse function - the arc tangent - and simplified, as in the previous step, the left side, output the final formula: α = arctg (a / b ).

Step 4:

If the well-known side of a right triangle - leg (a) and the hypotenuse (c), to calculate the angle (β), formed by these parties, use the inverse cosine, and she - the inverse cosine. Cosine is determined by the ratio of the length of the leg to the hypotenuse, and the formula in its final form can be written as: β = arccos (a / c). To calculate on the same raw data an acute angle (α), which lies opposite the famous leg, using the same ratio, replacing the arc cosine in the arcsine: α = arcsin (a / c).