1) The Topic of Trigonometry:
In geometry, if 3 elements out of 6 elements of a triangle (2 sides and 1 angle, 3 sides, or, 2 angles and 1 side) are known, the triangle can be drawn allowing us to measure its other 3 elements. Determining unknown sides and angles by measurement involves measurement errors and is approximate. To determine the unknown sides and angles exactly (by calculation), we need to develop some relations between sides and angles that can be used to solve for unknowns to any desired precision and accuracy. The determination of such relations and their related operations is the subject of Trigonometry.
Since length and angle are two different quantities, they can't be added or subtracted. To relate length and angle, we will define the "Trigonometric Ratios."
2) The Unit Circle:
|Any circle with a radius of 1 unit is called the "Unit Circle." As shown in the figure, Point A where the x-axis crosses the unit circle is chosen to be the Origin for selecting angles with counterclockwise as the positive direction. The unit circle may also be called the "Trigonometric Circle" for its use in Trigonometry.||
3) The Trigonometric Arc:
The trig. arc AM is an arc that starts from A, the Origin, and ends at M on the circle as shown. Now, this ending at M can happen in an infinite number of ways. One way to end at M is that an object or a point moves from A to M along arc AM sweeping an angle equal to θ. A second way is for the object or the point to start from A, complete 1 full turn arriving at A again, and then go to M. A third way is to complete 2 full turns arriving at A and then to M, and so on ....
Arc AM can therefore have the following values while ending at the same point M each time:
AM = (0)2π + θ or, AM = θ
AM = (1)2π + θ
AM = (2)2π + θ
AM = (3)2π + θ
AM = (k)2π + θ where k = 0,1,2,3,....∞
Note that k may take on negative values if the object or point moves in the negative direction (clockwise). The values of arc AM can therefore be
(-1)2π + θ , (-2)2π + θ .
(-3)2π + θ, and so on .... k = -1,-2,-3,...
4) The Sine, Cosine, Tangent, and Cotangent of an Angle:
We may define the following ratios on the basis of a Right Triangle. Note that sine and cosine are hypotenuse related, but tangent and cotangent are not.
For Angle B, for example, in the following figure, we define:
sinB is equal to the ratio of its opposite side b to Hypotenuse a,
cosB is equal to the ratio of its adjacent side c to Hypotenuse a,
tanB is equal to the ratio of its opposite side b to the adjacent side c, and
cotB is equal to the ratio of its adjacent side c to the opposite side b.
Note: The numerical values of the sine, cosine, tangent, and cotangent of any angle can be easily found buy using a scientific calculator. If an angle is given in degrees, the calculator must then be in degrees mode to calculate the correct values for the trig. ratios of that angle.
Example 1: In the following right triangle, find the unknown sides and angles:
Since the angles are in degrees, make sure that you have your calculator in "Degrees" mode. For any problem, in general, first cover the solution, draw the figure, and try to do it yourself. As soon as you solve for the numerical value of an unknown, write it down on the figure you drew until all unknowns are solved for. Look at the solution (not all the solution at once) step by step and make sure you understand the method.
Example 2: In the following right triangle, find the unknown sides and angles:
Example 3: In the following right triangle, assuming 3 significant figures on all numbers, find the unknown sides and angles:
Assuming 3 significant figures on all numbers, find the unknown sides or angles in each of the following triangles: