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Chapter
Functions of General Angles
Tables
Chapter 2: Trigonometryof
Solving Right Triangles
Law
Chapter 5: Vectors
INTRODUCTION
The word trigonometry comes from Greek words meaning measurement of triangles. Solving triangles is one of many aspects of trigonometry that you study today. To develop methods to solve triangles, trigonometric functions are constructed. The study of the properties of these functions and related applications form the subject matter of trigonometry. Trigonometry has applications in navigation, surveying, construction, and many other branches of science, including mathematics and physics.
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Chapter 1
TRIGONOMETRIC FUNCTIONS
Chapter Checkin
❑ Understanding angles and angle measurements
❑ Finding out about trigonometric functions of acute angles
❑ Defining trigonometric functions of general angles
❑ Using inverse notation and linear interpolation
Historically, trigonometry was developed to help find the measurements in triangles as an aid in navigation and surveying. Recently, trigonometry is used in numerous sciences to help explain natural phenomena. In this chapter, I define angle measure and basic trigonometric relationships and introduce the use of inverse trigonometric functions.
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 1-1a) result from counterclockwise rotation, and negative angles (Figure 1-1b) result from clockwise rotation. An angle with its initial side on the x-axis is said to be in standard position.
Figure 1-1 (a) A positive angle and (b) a negative angle.
Angles that are in standard position are said to be quadrantal if their terminal side coincides with a coordinate axis. Angles in standard position that are not quadrantal fall in one of the four quadrants, as shown in Figure 1-2.
Figure 1-2 Types of angles.
Third quadrant angle (a)
First quadrant angle (c)
Second quadrant angle (b)
° x y Fourth quadrant angle (d)
Quadrant angle (e)
Quadrant angle (f)
Example 1: Thefollowing angles (standard position) terminate in the listed quadrant.
94° 2nd quadrant
500° 2nd quadrant
–100° 3rd quadrant
180° quadrantal
–300° 1st quadrant
Two angles in standard position that share a common terminal side are said to be coterminal. The angles in Figure 1-3 are all coterminal with an angle that measures 30° .
All angles that are coterminal with d ° can be written as %%dn 360 $ + where n is an integer (positive, negative, or zero).
Example 2: Is an angle measuring 200° coterminal with an angle measuring 940°?
If an angle measuring 940° and an angle measuring 200° were coterminal, then
Because 740 is not a multiple of 360, these angles are not coterminal.
Figure 1-3 Angles coterminal with 70° .
Example 3: Name 4 angles that are coterminal with 70° .
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:
1 degree = 60 minutes
1 minute = 60 seconds or 1
Example 4: Write 34 15% l using decimal degrees.
Example 5: Write 12 18 44
using decimal degrees.
Example 6: Write 81. 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 xyr 222 += and refer to Figure 1-4.
1-4 Reference triangles.
side adjacent side opposite
Figure
sin sine of r y length of hypotenuse
length of side opposite ===aa a cos cosine of r x length of hypotenuse
length of side adjacent to ===aa a tan tangent of x y length of side adjacent length of side opposite ===aa a
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 cotangent, secant, and cosecant are trigonometric functions that are the reciprocals of the sine, cosine, and tangent, respectively.
csc cosecant of y r length of side opposite length of hypotenuse ===aa a sec secant of x r length of side adjacent to length of hypotenuse ===aa a cot cotangent of y x length of side opposite to length of side adjacent to ===aa a a
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.
Symbolically, (sin α)2 and sin2 α can be used interchangeably. From Figure 1-4 (a) and the Pythagorean theorem, xyr 222 += .
These three trigonometric identities are extremely important: cos sin tan = i i i
cos cot = i i i sin cos 1 22+=ii
Chapter 1: Trigonometric Functions
Example 7: Find sin θ and tan θ if θ is an acute angle <<0 90 i %% ak and cos θ = 1/4.
Example 8: Find sin θ and tan θ if θ is an acute angle <<0 90 i %% ak and 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 1-5, observe that XE and YE are complementary.
Figure 1-5 Reference triangles.
Thus, in general:
Example 9: What are the values of the six trigonometric functions for angles that measure 30°, 45°, and 60° (see Figure 1-6 and Table 1-1).
Figure 1-6 Drawings for Example 9.
Table 1-1Trigonometric Ratios for 30°, 45°, and 60° Angles
Functions of General Angles
Acute angles in standard position are all in the first quadrant, and all of their trigonometric functions exist and are positive in value. This is not necessarily true of angles in general. Some of the six trigonometric functions of quadrantal angles are undefined, and some of the six trigonometric functions have negative values, depending on the size of the angle. Angles in standard position have their terminal side in or between one of
the four quadrants. Figure 1-7 shows a point A (x, y) located on the terminal side of angle θ with r as the distance AO. Note that r is always positive. Based on the figures,
Figure 1-7 Positive angles in various quadrants.
(x,y)
(x,y)
(x,y)
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