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Book
Schaum's Outline of Trigonometry, 6th Edition

by Robert E. Moyer, The late Frank Ayres Jr.

Tough test questions? Missed lectures? Not enough time? Textbook too pricey?

Fortunately, there's Schaum's. This all-in-one-package includes hundreds of fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 detailed videos featuring math instructors who explain how to solve the most commonly tested problems—it's just like having your own virtual tutor! You'll find everything you need to build confidence, skills, and knowledge for the highest score possible.

More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. Helpful tables and illustrations increase your understanding of the subject at hand.

Schaum's Outline of Trigonometry, Sixth Edition, features:

• 620 practice problems with step-by-step solutions

• 218 supplementary problems to reinforce knowledge

• Concise explanations of all trigonometry concepts

• Updates that reflect the latest course scope and sequences, with coverage of periodic functions and curve graphing

• Support for all major textbooks for trigonometry courses

• Access to revised Schaums.com website with 20 problem-solving videos, and more

Book Chapter
1. Angles and Applications
Source – Schaum's Outline of Trigonometry, 6th Edition

1. Angles and Applications

Trigonometry is the branch of mathematics concerned with the measurement of the parts, sides, and angles of a triangle. Plane trigonometry, which is the topic of this book, is restricted to triangles lying in a plane. Trigonometry is based on certain ratios, called trigonometric functions, to be defined in the next chapter. The early applications of the trigonometric functions were to surveying, navigation, and engineering. These functions also play an important role in the study of all sorts of vibratory phenomena—sound, light, electricity, etc. As a consequence, a considerable portion of the subject matter is concerned with a study of the properties of and relations among the trigonometric functions.

Book Chapter
12. Area of a Triangle
Source – Schaum's Outline of Trigonometry, 6th Edition

12. Area of a Triangle

The area K of any triangle equals one-half the product of its base and altitude. In general, if enough information about a triangle is known so that it can be solved, then its area can be found.

Book Chapter
8. Basic Relationships and Identities
Source – Schaum's Outline of Trigonometry, 6th Edition

8. Basic Relationships and Identities

RECIPROCAL RELATIONSHIPS

QUOTIENT RELATIONSHIPS

PYTHAGOREAN RELATIONSHIPS

 csc θ= 1sin θ

tan θ=sin θcos θ

sin2 θ + cos2 θ = 1

sec θ=1cos θ

cotθ=cosθsinθ

1 + tan2 θ = sec2 θ

cotθ=1tanθ

1 + cot2 θ = csc2 θ

The basic relationships hold for every value of θ for which the functions involved are defined.

Thus, sin2 θ + cos2 θ = 1 holds for every value of θ, while tan θ = sin θ/cos θ holds for all values of θ for which tan θ is defined, i.e., for all θ ≠ n ⋅ 90° where n is odd. Note that for the excluded values of θ, cos θ = 0 and sin θ ≠ 0.

For proofs of the quotient and Pythagorean relationships, see Probs. 8.1 and 8.2. The reciprocal relationships were treated in Chap. 2.

(See also Probs. 8.3 to 8.6.)

Book Chapter
15. Complex Numbers
Source – Schaum's Outline of Trigonometry, 6th Edition

15. Complex Numbers

The square root of a negative number (e.g.,  −1,    −5, and  −9) is called an imaginary number. Since by definition  −5 = 5  · −1 and  −9 = 9· −1 = 3 −1, it is convenient to introduce the symbol i= −1 and to adopt  −5=i 5 and  −9=3i as the standard form for these numbers.

The symbol i has the property i2 = −1; and for higher integral powers we have i3 = i2 ⋅ i = (−1)i = −i, i4 = (i2)2 = (−1)2 = 1, i5 = i4 ⋅ i = i, etc.

The use of the standard form simplifies the operations on imaginary numbers and eliminates the possibility of certain common errors. Thus  −9·4= −36=6i since  −9  ·  4=3i(2)=6i but  −9  · −4   ≠ 36 since  −9  ·  −4=(3i)(2i)=6i2=−6.

Book Chapter
13. Inverses of Trigonometric Functions
Source – Schaum's Outline of Trigonometry, 6th Edition

13. Inverses of Trigonometric Functions

The equation

x=siny

(1)

defines a unique value of x for each given angle y. But when x is given, the equation may have no solution or many solutions. For example: if x = 2, there is no solution, since the sine of an angle never exceeds 1. If x=12, there are many solutions y = 30°, 150°, 390°, 510°, −210°, −330°, ....

y=arcsinx

(2)

In spite of the use of the word arc, (2) is to be interpreted as stating that "y is an angle whose sine is x." Similarly we shall write y = arccos x if x = cos y, y = arctan x if x = tan y, etc.

The notation y = sin−1 x, y = cos−1 x, etc. (to be read "inverse sine of x, inverse cosine of x," etc.) is also used but sin−1 x may be confused with 1/sin x = (sin x)−1, so care in writing negative exponents for trigonometric functions is needed.

Book Chapter
11. Oblique Triangles
Source – Schaum's Outline of Trigonometry, 6th Edition

11. Oblique Triangles

An oblique triangle is one which does not contain a right angle. Such a triangle contains either three acute angles or two acute angles and one obtuse angle.

The convention of denoting the angles by A, B, and C and the lengths of the corresponding opposite sides by a, b, and c will be used here. (See Fig. 11.1.)

Figure 11.1  
11x01

Book Chapter
5. Practical Applications
Source – Schaum's Outline of Trigonometry, 6th Edition

5. Practical Applications

The bearing of a point B from a point A in a horizontal plane is usually defined as the angle (always acute) made by the ray drawn from A through B with the north-south line through A. The bearing is then read from the north or south line toward the east or west. The angle used in expressing a bearing is usually stated in degrees and minutes. For example, see Fig. 5.1.

Figure 5.1  
05x01

In aeronautics, the bearing of B from A is more often given as the angle made by the ray AB with the north line through A, measured clockwise from the north (i.e., from the north around through the east). For example, see Fig. 5.2.

Figure 5.2  
05x02

Book Chapter
6. Reduction to Functions of Positive Acute Angles
Source – Schaum's Outline of Trigonometry, 6th Edition

6. Reduction to Functions of Positive Acute Angles

Let θ be any angle; then

sin(θ+n360°)=sinθ cot(θ+n360°)=cotθcos(θ+n360°)=cosθ sec(θ+n360°)=secθtan(θ+n360°)=tanθ csc(θ+n360°)=cscθ

where n is any integer (positive, negative, or zero).

EXAMPLE 6.1

  1. sin 400° = sin(40° + 360°) = sin 40°

  2. cos 850° = cos(130° + 2 ⋅ 360°) = cos 130°

  3. tan(−1000°) = tan(80° − 3 ⋅ 360°) = tan 80°

If x is an angle in radian measure, then

sin(x+2nπ)=sinx cot(x+2nπ)=cotxcos(x+2nπ)=cosx sec(x+2nπ)=secxtan(x+2nπ)=tanx csc(x+2nπ)=cscx

where n is any integer.

EXAMPLE 6.2

  1. sin 11π/5 = sin(π/5 + 2π) = sin π/5

  2. cos(−27π/11) = cos[17π/11 − 2(2π)] = cos 17π/11

  3. tan 137π = tan[π + 68(2π)] = tan π

Book Chapter
4. Solution of Right Triangles
Source – Schaum's Outline of Trigonometry, 6th Edition

4. Solution of Right Triangles

The solution of right triangles depends on using approximate values for trigonometric functions of acute angles. An important part of the solution is determining the appropriate value to use for a trigonometric function. This part of the solution is different when you are using tables (as in Secs. 4.2 to 4.4) from when you are using a scientific calculator (as in Secs. 4.5 and 4.6.)

In general, the procedure will be to use the given data to write an equation using a trigonometric function and then to solve for the unknown value in the equation. The given data will consist either of two sides of a right triangle or of one side and an acute angle. Once one value has been found, a second acute angle and the remaining side can be found. The second acute angle is found using the fact that the acute angles of a right triangle are complementary (add up to 90°). The third side is found by using a definition of a second trigonometric function or by using the Pythagorean theorem (see App. 1, Geometry).

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