TRIGONOMETRY III

Trigonometric ratios of acute angles

Trigonometric ratios in a right-angled triangle

Trigonometric ratios of an acute angle are defined in terms of the sides of the triangle and are independent of its size.  The trigonometric ratios sine, cosine and tangent of the acute angle in a right angled triangle like that in the figure, in which the angle B=90º, b is the hypotenuse, and a and c are the sides of the right angle, are difined like this:

sin A=a/b, cos A=c/b, tg A=a/c.

If the length of the sides of the triangle ABC is increased by extending the sides b and c and if parallel straight lines to the side a are drawn, then similar triangles to the original one are obtained and, therefore, the trigonometric ratios of the angle A are still the same, and depend only on the amplitude of A (in degrees or radians).

Vary the value of b until it reaches a length of 12. Observe how the values of the trigonometric ratios of the angle of 30º that appear in the figure don't vary.  Change to 45º and 60º using Clear. Calculate the trigonometric ratios of the angles of 15º, 1 radian, 85º and 0.3 radians.

A triangle of hypotenuse of length one

Given that the value of the trigometric ratios in a right angled triangle doesn't depend  on the length of the sides, a triangle can be chosen whose hypotenuse is b=1. In this case the calculations will be a lot more simple.

Repeat in this case the calculation of the trigonometric ratios of the angles 15º, 45º, 60º, 1 radian, 85º and 0.3 radians. Do you find any relationship between the three trigonometric ratios?  Try to write a formula that relates them.

Trigonometric ratios of complementary angles

As the sum of the angles of any triangle is 180º, in a right angled triangle, the two acute angles must add up to 90º.  It is also said that two angles which add up to 90º are complementary.  The Descartes applet below shows simultaneously the values of the trigonometric ratios of the angles A and C, which as we already know are complementary.

Do you find any relationship between the trigonometric ratios?  Check with A: 45º, 60º, 33.4º, 72º and 85,7º. Are the previous relationships between the ratios of A and C maintained?  If sin A=0.391 and cos A=0.921 are known, would you know how to calculate tan A and the three trigonometric ratios of the complementary angle of A?

Author: Miguel García Reyes

	Ministerio de Educación, Cultura y Deporte. Año 2000