24 Nisan 2011 Pazar

Part 1: Trigonometry

Angles

An angle is a measure of rotation. Angles are measured in degrees. One complete rotation is measured as 360°. Angle measure can be positive or negative, depending on the direction of rotation. The angle measure is the amount of rotation between the two rays forming the angle. Rotation is measured from the initial side to the terminal side of the angle. Positive angles (Figure 1 a) result from counterclockwise rotation, and negative angles (Figure 1 b) result from clockwise rotation. An angle with its initial side on the x-axis is said to be instandard position.






Figure 1
(a) A positive angle and (b) a negative angle.




Figure 2
Types of angles.


94°
2nd quadrant
500°
2nd quadrant
−100°
3rd quadrant
180°
quadrantal
−300°
1st quadrant





Figure 3
Angles coterminal with 30°









Because 740 is not a multiple of 360, these angles are not coterminal.
Example 3: Name 5 angles that are coterminal with −70°.




Angle measurements are not always whole numbers. Fractional degree measure can be expressed either as a decimal part of a degree, such as 34.25°, or by using standard divisions of a degree called minutes and seconds. The following relationships exist between degrees, minutes, and seconds:




Example 4: Write 34°15′ using decimal degrees.




Example 5: Write 12°18′44″ using decimal degrees.




Example 6: Write 81.293° using degrees, minutes, and seconds.




Functions of Acute Angles

The characteristics of similar triangles, originally formulated by Euclid, are the building blocks of trigonometry. Euclid's theorems state if two angles of one triangle have the same measure as two angles of another triangle, then the two triangles are similar. Also, in similar triangles, angle measure and ratios of corresponding sides are preserved. Because all right triangles contain a 90° angle, all right triangles that contain another angle of equal measure must be similar. Therefore, the ratio of the corresponding sides of these triangles must be equal in value. These relationships lead to the trigonometric ratios. Lowercase Greek letters are usually used to name angle measures. It doesn't matter which letter is used, but two that are used quite often are alpha (α) and theta (θ).


Angles can be measured in one of two units: degrees or radians. The relationship between these two measures may be expressed as follows: 




The following ratios are defined using a circle with the equation x2 + y2 = r2 and refer to Figure1 .






Figure 1
Reference triangles.





Remember, if the angles of a triangle remain the same, but the sides increase or decrease in length proportionally, these ratios remain the same. Therefore, trigonometric ratios in right triangles are dependent only on the size of the angles, not on the lengths of the sides.
The cosecant, secant, and cotangent are trigonometric functions that are the reciprocals of the sine, cosine, and tangent, respectively.




If trigonometric functions of an angle θ are combined in an equation and the equation is valid for all values of θ, then the equation is known as a trigonometric identity. Using the trigonometric ratios shown in the preceding equation, the following trigonometric identities can be constructed.








These three trigonometric identities are extremely important: 



Example 1: Find sin θ and tan θ if θ is an acute angle (0° ≤ θ ≤ 90°) and cos θ = ¼.




Example 2: Find sin θ and cos θ if θ is an acute angle (0° ≤ θ ≤ 90°) tan θ = 6.
If the tangent of an angle is 6, then the ratio of the side opposite the angle and the side adjacent to the angle is 6. Because all right triangles with this ratio are similar, the hypotenuse can be found by choosing 1 and 6 as the values of the two legs of the right triangle and then applying the Pythagorean theorem.




Trigonometric functions come in three pairs that are referred to as cofunctions. The sine and cosine are cofunctions. The tangent and cotangent are cofunctions. The secant and cosecant are cofunctions. From right triangle XYZ, the following identities can be derived:








Using Figure 2 , observe that ∠X and ∠Y are complementary.





Figure 2
Reference triangles.

Thus, in general: 



TABLE 1Trigonometric Ratios for 30°, 45°, and 60° Angles








Figure 3
Drawings for Example 3.

Functions of General Angles













Figure 1
Positive angles in various quadrants.

Law of Cosines












Figure 1
Reference triangle for Law of Cosines.

From the figure,




Thus the coordinates of A are




Remember, all three forms of the Law of Cosines are true even if γ is acute. Using the distance formula,




In the preceding formula, if γ is 90°, then the cos 90° = 0, yielding the Pythagorean theorem for right triangles. If the orientation of the triangle is changed to have A or B at the origin, then the other two versions of the Law of Cosines can be obtained.
Two specific cases are of particular importance. First, use the Law of Cosines to solve a triangle if the length of the three sides is known.
Example 1: If α, β, and γ are the angles of a triangle, and ab, and c are the lengths of the three sides opposite α, β, and γ, respectively, and a = 12, b = 7, and c = 6, then find the measure of β.
Use the form of the Law of Cosines that uses the angle in question.




Rewrite solving for cos β.




Because cos β > 0 and β < 180°, β < 90° thus,




The measure of α can be found in a similar way.




Rewrite solving for cos α.




Because cos α < 0 and α 180° α > 90° Thus,




Because the three angles of the triangle must add up to 180°,



Law of Sines








Figure 1
Reference triangles for Law of Sines.







In Figure 1 (b), ∠CBD has the same measure as the reference angle for β Thus,




It follows that




Similarly, if an altitude is drawn from A,




Combining the preceding two results yields what is known as the Law of Sines.




In other words, in any given triangle, the ratio of the length of a side and the sine of the angle opposite that side is a constant. The Law of Sines is valid for obtuse triangles as well as acute and right triangles, because the value of the sine is positive in both the first and second quadrant—that is, for angles less than 180°. You can use this relationship to solve triangles given the length of a side and the measure of two angles, or given the lengths of two sides and one opposite angle. (Remember that the Law of Cosines is used to solve triangles given other configurations of known sides and angles).
First, consider using the Law of Sines to solve a triangle given two angles and one side.
Example 1: Solve the triangle in Figure 2 given θ = 32°, χ = 77°, and d = 12.





Figure 2
Drawing for Example 1.







In fact,

Areas of Triangles

The most common formula for finding the area of a triangle is K = ½ bh, where K is the area of the triangle, b is the base of the triangle, and h is the height. (The letter K is used for the area of the triangle to avoid confusion when using the letter A to name an angle of a triangle.) Three additional categories of area formulas are useful.
wo sides and the included angle (SAS): Given Δ ABC (Figure 1 ), the height is given by h = c sinA. Therefore,





Figure 1
Reference triangles for area formulas.








Similarly,











Introduction to Graphs

You can measure angle sizes by using more than one scale. The degree scale is probably the most well-known scale, although the radian scale is equally as popular and useful. Although most applications deal only in one of these two scales, it is important to understand their differences and how to convert from one to the other.



You can define trig functions by using a unit circle, a circle with a radius of 1. As a point revolves around the circle, its distance from the x-axis is defined as the sine, and its distance from the y-axis is defined as the cosine. These definitions match the previous definitions in terms of a right triangle. The graphs of trigonometric functions are used to visually represent their behavior.


Periodic and Symmetric Functions

The unit circle has a circumference of C = 2π r = 2π(1) = 2π. Therefore, if a point P travels around the unit circle for a distance of 2π, it ends up where it started. In other words, for any given value q, if 2π is added or subtracted, the coordinates of point P remain unchanged (Figure 1 ).






Figure 1
Periodic coterminal angles.



It follows that 




If k is an integer,




Functions that have this property are called periodic functions. A function f is periodic if there is a positive real number q such that fx + q) = fx) for all x in the domain of f. The smallest possible value for q for which this is true is called the period of f.
Example 1: If sin y = y = (3/5)/10, then what is the value of each of the following: sin( y + 8π), sin( y + 6π), ( y + 210π)?
All three have the same value of  because the sine function is periodic and has a period of 2π.
The study of the periodic properties of circular functions leads to solutions of many real-world problems. These problems include planetary motion, sound waves, electric current generation, earthquake waves, and tide movements.
Example 2: The graph in Figure 2 represents a function f that has a period of 4. What would the graph look like for the interval −10 ⩽ x ⩽ 10?





Figure 2
Drawing for Example 2.





Figure 3
Drawing for Example 2.










Figure 4
Even and odd trig functions.

The cosine is known as an even function, and the sine is known as an odd function. Generally speaking,




for every value of x in the domain of g. Some functions are odd, some are even, and some are neither odd nor even.
If a function is even, then the graph of the function will be symmetric with the y-axis. Alternatively, for every point on the graph, the point (− x, − y) will also be on the graph.
If a function is odd, then the graph of the function will be symmetric with the origin. Alternatively, for every point ( xy) on the graph, the point (− x, − y) will also be on the graph.




Figure 5
Drawings for Example 3.





Figure 6
Drawings for Example 4.

Example 5: Is the function f(x) = 2 x3 + x even, odd, or neither? 



Because f(−x) = − f(x), the function is odd.
Example 6: Is the function f(x) = sin x – cos x even, odd, or neither?




the function is neither even nor odd. Note: The sum of an odd function and an even function is neither even nor odd.
Example 7: Is the function fx) = x sin x cos x even, odd, or neither?




Because f(− x) = fx), the function is even.





Graphs: Sine and Cosine

To see how the sine and cosine functions are graphed, use a calculator, a computer, or a set of trigonometry tables to determine the values of the sine and cosine functions for a number of different degree (or radian) measures (see Table 1 ).
TABLE 1Values of the Sine and Cosine at Various Angles
degrees
30°
45°
60°
90°
120°
radians
0
sin x
0
0.500
0.707
0.866
1
0.866
cos x
1
0.866
0.707
0.500
0
−0.500
degrees
135°
150°
180°
210°
225°
240°
radians
π
sin x
0.707
0.500
0
−0.500
−0.707
−0.866
cos x
−0.707
−0.866
−1
−0.866
−0.707
−0.500








Figure 1
One period of the a) sine function and b) cosine function.






Figure 2
Multiple periods of the a) sine function and b) cosine function.

Several additional terms and factors can be added to the sine and cosine functions, which modify their shapes.




Figure 3
Examples of several vertical shifts of the sine function.





Figure 4
Examples of several amplitudes of the sine function.

Combining these figures yields the functions y = A + B sin x and also y = A + B cos x. These two functions have minimum and maximum values as defined by the following formulas. The maximum value of the function is M = A + |B|. This maximum value occurs whenever sin x = 1 or cos x = 1. The minimum value of the function is m = A - |B|. This minimum occurs whenever sin x= −1 or cos x = −1.
Example 1: Graph the function y = 1 + 2 sin x. What are the maximum and minimum values of the function?




Figure 5
Drawing for Example 1.

Example 2: Graph the function y = 4 + 3 sin x. What are the maximum and minimum values of the function?




Figure 6
Drawing for Example 2.





Figure 7
Examples of several frequencies of the a) sine function and b) cosine function.











Figure 8
Examples of several phase shifts of the sine function.

Example 3: What is the amplitude, period, phase shift, maximum, and minimum values of
  1. y = 3+2 sin (3 x-2)
  2. y = 4 cos2π x
TABLE 2Attributes of the General Sine Function
Function
Amplitude
Period
Phase Shift
Maximum
Minimum
y = 3 + 2 sin (3 x - 2)
2
2 (right)
5
1
2 (right)
y = 4 cos 2π x
4
1
0
4
−4
Example 4: Sketch the graph of y = cosπ x.




Figure 9
Drawing for Example 4.

Example 5: Sketch the graph of y = 3 cos (2x + π/2).




Figure 10
Drawing for Example 5.





It is important to understand the relationships between the sine and cosine functions and how phase shifts can alter their graphs.


Graphs: Other Trigonometric Functions

The tangent is an odd function because,




The tangent has a period of π because




The tangent is undefined whenever cos x = 0. This occurs when x = qπ/2, where q is an odd integer. At these points, the value of the tangent approaches infinity and is undefined. When graphing the tangent, a dashed line is used to show where the value of the tangent is undefined. These lines are called asymptotes. The values of the tangent for various angle sizes are shown in Table 1 .
TABLE 1Values of the Tangent Function at Various Angles
degrees
30°
45°
60°
75°
80°
85°
87°
90°
radians
0
tan x
0
0.577
1
1.73
3.73
5.67
11.43
19.08






Figure 1
A portion of the tangent function.



The tangent is an odd function and is symmetric about the origin. The graph of the tangent over several periods is shown in Figure 2 . Note that the asymptotes are shown as dashed lines, and the value of the tangent is undefined at these points.





Figure 2
Several periods of the tangent function.



The cotangent is the reciprocal of the tangent, and its graph is shown in Figure 3 . Note the difference between the graph of the tangent and the cotangent in the interval from 0 to π/2.





Figure 3
A portion of the cotangent function.



As shown in Figure 4 , in the graph of the cotangent, the asymptotes are located at multiples of π.





Figure 4
Several periods of the cotangent function.









and graph at least two complete periods of the function.
The asymptotes can be found by solving Cx + D = π/2 and Cx + D = −π/2 for X.




The period of the function is




The phase shift of the function is









Figure 5
Phase shift of the tangent function.





and graph at least two periods of the function.




The period of the function is 



The phase shift of the function is 



Because the phase shift is positive, it is to the left.
The graph of the reciprocal function 



is shown in Figure 
6 . Graphing the sine (or cosine) can make it easier to graph the cosecant (or secant).






Figure 6
Several periods of the cosecant function and the sine function.



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