Calculation of right-angled triangles

Triangle with right angle

Task 1.: In a right-angled triangle, the hypotenuse c = 4 cm and the angle = 50º. Find a and b, rounded to 4 digits!

sin= a/c		a = c·sin		cos = b/c		b = c cos
a = 4·0.7660		a = 3.064 cm		b = 4·0.6428		b = 2.571

A triangle with one right angle is fully determined by two items, whence follow a number of tasks for the calculation of the other sides and angles.

All these problems can be solved, because every trigonometric function interlinks one angle and two sides, that is three items, one of which can be found if two are given. All possibilities of interlinkage are given by the trigonometric functions; you only need select the correct function! In this context, Pythagoras' Theorem, which interlinks the three sides, is frequently employed. Often there are several possible ways to obtain the solution.

One partial triangle has a right angle

If the given triangle does not have a right angle, since the trigonometric functions can only be applied to right-angled triangles, you have to try to create them.

Task: Analyze the isosceles triangle ABC when c and g are given.

Draw the height from C to AB and create the right-angled triangles ADC and DBC.

Always take the geometrical properties of figures into account!

In this triangle, = b, CD bisects g as well as c, whence b = a= 90º - g/2,
sin g/2 = c/2:a or a = c/2·sin g/2 = b.

Caution! Do not cancel the 2 in the expression for a!

Task : In a rectangle are given b and e. Find a!

The rectangle has the properties of parallelograms and, moreover, its diagonals are equal. The triangle MBC is isosceles and is decomposed into two right-angled triangles by the height ME to CB: ME = a/2. The triangle ABC is right-angled, etc. Often you can solve a problem in different ways. Because MBC is isosceles, the angle ACB equals 90º - e /2.

Task : You are given the diagonals e and f of a rhombus. Find a!

A rhombus has the properties of parallelograms; in addition, its diagonals are perpendicular to each other and bisect its angles. The right-angled triangle AMB yields

tan a/2 = (f/2) : (e/2) or tan a/2 = f / e. With a/2 you also know a.

Task 4: You are to place a saddle roof on a house with a rectangular ground plan. Compute the size of the two faces of the roof ABFE and DCFE, which are to be inclined by 20º with respect to the horizontal plane! AC = a = 10 m, BC = b = 6 m. The triangle BCF is isosceles, whence
cos a = b/2 : s or s = b/2cos a. The area D of each of the two rectangular roof faces can be computed from AB · BF = a · s:

D = a·s = a · b/2 cos a = 10·6/2 cos 20 = 31.9.

Each of the faces of the roof faces has 31.9 m².

More accurate computation. Interpolation

If your table does not present exactly a value of a trigonometric function, which you require, we have so far rounded off; if you want more exact values, you must interpolate. However, it does not make sense to exceed the number of digits given in your table! You can transfer the method, which you know from the discussion of logarithms, with the one difference that some of the trigonometric functions for larger a can increase or decrease rapidly:

sin 0º = 0		tan 0º = 0		cos 0º = 1		cot 0º =
sin 90º = 1		tan 90º =		cos 90º = 0		cot 90º = 0

so that, as the angles increase, sin and tan become larger, cos and cot smaller, as shown by the table.

Task: Find sin 35.78.

sin increases as the angle increases. Assuming uniform growth in the small interval. you have for 10 units in the last decimal 15 units of the last decimal in sin, that is for 8 units of the angle 12 units of sin. By such a final calculation or through proportions or through the table of proportions in your tables, you can find the required value. The proportion would be 10/8 = 15/x, whence x = 15·8/10 = 12.

Task: Find cos 35.78!

cos decreases as the angle increases. Assuming a uniform decrease in the small angle interval of 10 units in the last decimal, cos decreases by 8 units in the last decimal.

Task: Given cot a = 1.388, find a.

As cot decreases by 4 units in the last decimal, the angle increases, assuming uniform growth, by 8 in the last decimal. The proportion is 5/4 = 10/x.

For logarithmic computations, you must first find for a given the value of the trigonometric function. In order to simplify matters, your table also presents logarithms of the trigonometric functions.

Task: Find log cos 35.78!

In order to simplify matters, your table uses instead of 0.···· -1, 1.···· -2, etc. 9.···· -10 or 8. ··· - 10.

Task: Find the angle a, given log cos a = 9.8870 - 10!

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