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Trigonometric Ratios: Basic Trigonometry depends on similar triangles. Because the triangles are similar (corresponding sides are in the ratio 1:2) the two triangles will have the same angles. In the case of the 3,4,5 triangle, shown below, the angle A is about 37 degrees. One of the consequences of the similarity of these two triangles (and all other right angled trinagles that contain 37 degrees) is that the ratios of pairs of sides within the same triangle will all be the same. If, for example, we look at the ratio of the vertical side (the side opposite angle A) to the hypotenuse of the triangle, the result is 3:5 or 3/5 = 0.6 for the smaller triangle and 6:10 or 6/10 =0.6 for the larger. The same result will be obtained for all right angled triangles that contain angel A and this ratio is known as the sine of angle and can be found using a calculator. The ratios between the lenghts of the sides of a right angeled triangle are known as the trigonometric ratios.
The most commonly used method of remembering the trigonometric ratios is to label the sides of the triangle in the following way.
The hypotenuse is the side opposite the right angle. In all basic trgonometric problems one of the othere angels will be important (either it is known or needs to be found). The side oppostie this angle is known as the oppostie side. The third side that is next to this angle is known as the adjacent side. The main trigonometric functions are now defined as follows:
sin X = opp/hyp cos X = adj/hyp tan X = opp/adj
These definitions are the basis for solving right angled triangles and have a number of applications including navigation and surveying.
Let's talk about the SINE RULE....
The Sine Rule can be stated as: a/sin A = b/sin B = c/sin C
or
sin A/a = sin B/b = sin C/c
~EXAMPLE~
Solve the triangle given the lengths of the sides in cenitmeters, correct to one decimal place and angle correct to the nearest degree.
a=23.8cm
Theta (angle A)= 83 degrees
Alpha (angle B)= 47 degrees
Since two of the angles are known, the third is: C = 180-47-83 = 50 degrees The lengths of the remaining sides must be found using the known pairing of side and angle, b and B. a/sin A = b/sin B
a/sin 47 = 23.8/sin 83
a = 23.8 x sin 47/sin 83
a = 17.5 cm
similarly, the remaining side can be calculated: c/sin C = b/sin B
c/sin 50 = 23.8/sin 83
a = 23.8 x sin 50/sin 83
a = 18.4 cm
Now let's look at the COSINE RULE....
The Cosine Rule is usually applied to triangles in which we do not know a pairing of an angle and the opposite side. It is generally harder to use and should not be used if simple trigonometry or the Sine Rule can be used instead. The Cosine Rule, with the standard labeling of the triangle has three versions
a^2 = b^2 + c^2 - 2bc cos A
b^2 = a^2 + c^2 - 2ac cos B
c^2 = a^2 + b^2 -2ab cos C
~EXAMPLE~
Solve the following triangle giving the length of the side in centimeters and angles to the nearest degree.\
a = 10.5 cm
b = 6 cm
C = 69 degrees
The solution is:
c^2 = a^2 + b^2 - 2ab cos C
c^2 = 10.5^2 + 6^2 - 2 x 10.5 x 6 x cos 69
c^2 = 101.1 cm
The remaining angles can be calculated using the Sine Rule. (see above)
Final Answer: B = 34 degrees A = 77 degrees
Well, thanks for visiting our site! Hope you learned a lot about trigonometry.
Love, Nick, Robyn, and Avery
