4.2 Trigonometric
Functions
The six trigonometric functions are traditionally defined in terms of the six ratios of the sides of a right triangle. This approach is valid for positive angles of measure smaller than 90o. We will look at the traditional approach first, then we will look at a method of defining the trigonometric functions for angles of either sign and of any size.
Consider a right triangle ABC with the angle C being the 90o angle.
We will allow the letters A, B and C do double duty. They will represent the vertices of the triangle, as well as the interior angles whose initial and terminal sides meet at that vertex.
Then we can define the six trigonometric functions of the angle A in terms of the six ratios of the triangle.
The sine function: sin A = BC / AB
The cosine function: cos A = AC / AB
The tangent function: tan A = BC / AC= sin A / cos A
The cosecant function: csc A = AB / BC = 1 / sin A
The secant function: sec A = AB / AC = 1 / cos A
The cotangent function: cot A = AC / BC = 1 / tan A = cos A / sin A
Exercise 4.2.1
Suppose AC = 3, BC = 2. Find all six trigonometric functions of A. [Hint: You’ll have to use the Pythagorean Theorem to find AB.] Find all six trigonometric functions of B. B is called the complement of A, since any two positive angles which sum to 90o are called complementary angles. How does sin A compare to cos B?
How does tan A compare to cot B? How does sec A compare to csc B? Do you see why the ‘co’ in cosine, cosecant and cotangent refers to ‘complement’ ?
There are special triangles which allow us to find the trigonometric functions of 30 o, 45 o, and 60 o.
The right triangle with AC = 1 and BC = 1 is the part of the unit square lying below the diagonal AB, with AB calculated using the Pythagorean theorem.
Exercise 4.2.2
Use the special triangle mentioned above to find all six
trigonometric functions of the angle 45 o.
The right triangle with AC = 1 and AB = 2 is half of the equilateral triangle having all sides equal to 2. It can be used to find all six trigonometric functions of either 60o or 30o.
Exercise 4.2.3
Find all six trigonometric functions of A = 60o
Exercise 4.2.4
Find all six trigonometric functions of A = 30o
Angles which are multiples of 30o and 45o are called special angles. When exercises employ the trigonometric functions of special angles, exact values are required. Decimal approximations are not acceptable.
In order to find the trigonometric functions of angles which are not between 0o and 90o in size, we place the angle in standard position with the vertex at the origin and the initial side coinciding with the positive x-axis. Then pick any point ( x, y ) on the terminal side of the angle. Let r denote the distance of the point ( x, y ) from the origin. Then r2 = x2 + y2. The value of x or of y may be negative, but the value of r will never be negative. The six trigonometric functions may then be defined as follows:
The sine function: sin A = y / r
The cosine function: cos A = x / r
The tangent function: tan A = y / x = sin A / cos A
The cosecant function: csc A = r / y = 1 / sin A
The secant function: sec A = r / x = 1 / cos A
The cotangent function: cot A = x / y = 1 / tan A = A = cos A / sin A
Exercise 4.2.5
The terminal side of a standard angle A passes through the point ( -1,2 ). Find all six trigonometric functions of the angle. You will have to find the value of r first.
Notice that some of the six trigonometric functions have negative values in the second quadrant, and some of them have positive values. This is because x and y are negative is some quadrants, but not others.
Exercise 4.2.6
Determine the sign of x and y in each of the four quadrants. Use the definitions of the six trigonometric functions (and the fact that r is never negative) to determine the sign of the six trigonometric functions in each of the four quadrants.
A quadrant angle is an angle in standard position whose terminal side lies on one of the axes. For example,
0o, 90o, 180o, 270o are quadrant angles. Others would be 360o, 450o, -90o, -180o, -270o, etc. Some of the trigonometric functions are undefined for some of the quadrant angles.
Exercise 4.2.7
Use the points (0,1), (1,0), (-1,0) and (0,-1) as terminal points on 0o, 90o, 180o, and 270o respectively in order to find all six trigonometric values of the angles. Put the results into a table, indicating which trigonometric functions are undefined for which of the quadrant angles.
The process of picking a point on the terminal side of an angle in standard position can be simplified if we agree to always pick the point exactly one unit from the origin so that r will always equal 1. In other words: Always pick ( x, y ) so that it lies on the unit circle. When we do this we get an interesting and useful result:
If A is an angle in standard position crossing the unit
circle at ( x, y ), then x = cos A and y = sin A.
Since the equation of the unit circle is x 2 + y 2 = 1, this implies another useful result:
( cos A ) 2
+ ( sin A ) 2 = 1
Exercise 4.2.8
Sketch the coordinate axis and the graph of the unit circle. Draw an angle A in standard position and with terminal side in the first quadrant. Draw the angle – A on the same diagram. Note the positions where A and
– A cross the unit circle. Label the point where the terminal side of A crosses the circle as ( cos A, sin A ) and the point where the terminal side of – A crosses the circle as ( cos ( - A ), sin ( - A ) ). Can you guess the relationship between cos ( - A ) and cos A ? Between sin ( - A ) and sin A ? Recall that a function f is an odd function of f ( - x ) = - f ( x ) and is even if f ( - x ) = f ( x ). Is the cosine function even or is it odd? Is the sine function even or is it odd? What about the tangent function? Use the fact that tan A = sin A / cos A .
Exercise 4.2.9
Sketch the coordinate axes and a large unit circle almost as wide as the sheet of paper. Use a compass to draw the circle. Without changing the radius of the compass, place the point of the compass at the point (1,0) and mark a point on the circle in quadrant I and in quadrant IV with the pen of the compass. Then place the point of the compass at the point (0,1) and mark points on the circle in quadrants I and II. For each of the four points you marked on the circle, do the following: Draw a line through the point and the center of the circle, continuing the line until it crosses the opposite side of the circle. When you have finished, the circle should be divided into 12 equal pie shaped wedges. Now, beginning with the point (1,0) and moving counterclockwise around the circle, label each of the twelve points with the angle from 0 o to 330 o whose terminal side crosses at that point. Label the points a second time using radian measure.
Exercise 4.2.10
Using the diagram you drew in exercise 4.2.9, label the points you drew on the circle in quadrant I with their coordinates. Remember x = cos A and y = sin A. Now label the coordinates of the quadrant angles. Then label the coordinates of all the remaining points on the circle using symmetry. They will have the same coordinates of the points in quadrant I except that the signs of some coordinates will be different depending upon the quadrant.
Exercise 4.2.11
Sketch the coordinate axes and a large unit circle almost as wide as the sheet of paper. Use a compass to draw the circle. Draw the terminal side of the angles 45 o, 135 o, 225 o and 315 o. This should divide the circle into eight equal pie shaped wedges. Label the coordinates of each of these eight points.
With the aid of the two previous diagrams, you should be able to find the sine and cosine of any of the special angles—the angles which are multiples of 30o and 45o.
Exercise 4.2.12
Using a compass draw a large unit circle on a sheet of paper. Draw the coordinate axes centered at the center of the circle. Mark a point P on the circle in quadrant I and draw the terminal side of the angle A which crosses the circle at P. Draw a horizontal and through P until it intersects the circle in quadrant II . Label that intersection point P’. Draw a vertical line through P’ until it crosses the circle in quadrant III. Label that point P”. Draw a horizontal line through P” until it crosses the circle in quadrant IV. Label that point P”’. Finally, draw a vertical line through the point P”’. If you were very careful, this last vertical line should cross the circle exactly at the original point P, and you should have produced a rectangle through the four points. Now draw the terminal sides of angles A’, A”, and A”’ through the corners of the rectangle. Label the coordinates of P as ( x, y ), the coordinates of P’ as ( - x, y ), the coordinates of P” as ( - x, - y ) and the coordinates of P”’ as ( ? ).
Notice the implication. The sine and cosine functions (and therefore all the trigonometric function) of A’, A” and A”’ can be deduced from the corresponding sine and cosine functions of A. It will only be necessary to find the proper sign to affix, depending upon the quadrant in which the angle lies.
The rectangle you drew in exercise 4.2.12 is called the reference rectangle, of each of the angles passing through its vertices, and the angle passing through the vertex in quadrant I is the reference angle of the other three angles. The reference angle will always be between 0 o and 90 o.
Exercise 4.2.13
Find the reference angles of each of the following angles:
(a) 112 o (b) 195 o (c) 295 o (d) 395 o (e) - 295 o