• Trigonometry involves the measurement of triangles. Trigonometry includes the measurement of angles, lengths, and arc lengths of triangles in circles and planes and also in spheres. Trigonometry is used in engineering, navigation, the study of electricity, light and sound, and in any field involving the study of periodic and wave properties.

• Trigonometric functions can be defined using ratios of sides of a right triangle and, more generally, using the coordinates of points on a circle of radius one. Trigonometric functions are sometimes called circular functions because their domains are lengths of arcs on a circle. Sine, cosine, tangent, cotangent, secant, and cosecant are trigonometric functions. For example, for a circle having a radius of one:

Point P has coordinates (cos x, sin x), and the arc distance of the angle has x units of length and is measured in radians.

• In this chapter trigonometric functions, identities, definitions, graphs, and relationships are presented.

• Certain trigonometric functions describe triangles formed in coordinate systems. For a rectangular coordinate system with an X axis and Y axis, an angle with its vertex at x = 0, y = 0 can be drawn. An angle is said to be in “standard position” if its vertex is at (0, 0) of the X-Y coordinate system and if one side lies on the positive side of the X axis.

• If the standard position angle is measured in a counterclockwise direction, it is positive. If the standard position angle is measured in a clockwise direction, it is negative.

• The angle, depending on the direction it is measured, has an “initial side” where its measurement begins and a “terminal side” where its measurement ends.

• The following four figures are examples of standard position angles with their terminal sides in the upper right, upper left, lower right, and lower left quadrants.

Standard Position Angle, Ø

Standard Position Angle, Ø

Standard Position Angle, Ø

Standard Position Angle, Ø

• Trigonometric functions defined in terms of the angle (Ø), x, y, and radius r are:

sine Ø = sin Ø = y/r

cosine Ø = cos Ø = x/r

tangent Ø = tan Ø = y/x

cosecant Ø = csc Ø = r/y

secant Ø = sec Ø = r/x

cotangent Ø = cot Ø = x/y

• Trigonometric identities are:

cos Ø = 1/sec Ø

tan Ø = 1/cot Ø

sec Ø = 1/cos Ø

cot Ø = 1/tan Ø

cos Ø tan Ø = sin Ø

1 + tan² Ø = sec² Ø

sin² Ø + cos² Ø = 1

sin Ø = 1/csc Ø

tan Ø = sin Ø/cos Ø

csc Ø = 1/sin Ø

cot Ø = cos Ø/sin Ø

cos Ø csc Ø = cot Ø

1 + cot² Ø = csc² Ø

• Trigonometric functions can be defined by considering the right triangle formed. If r is the hypotenuse, y is the side opposite Ø, and x is the side adjacent to Ø, then the following is true:

sin Ø = opposite/hypotenuse = y/r

csc Ø = hypotenuse/opposite = r/y

cos Ø = adjacent/hypotenuse = x/r

sec Ø = hypotenuse/adjacent = r/x

tan Ø = opposite/adjacent = y/x

cot Ø = adjacent/opposite = x/y

(y/r)² + (x/r)² = 1 or y² + x² = r²

Note that the Pythagorean Theorem is r² = x² + y²

(To remember sin Ø, cos Ø, and tan Ø, think of the word SohCahToa or S^o_hC^a_hT^o_a or S^o/_hC^a/_hT^o/_a.)

• Examples of trigonometric functions, (π ≈ 3.14):

(See a trigonometry book, calculus book, or book of mathematical tables for more values.)

• Trigonometric functions of angles of a 30:60:90 triangle.

(a = adjacent, o = opposite, h = hypotenuse)

30 degrees = π/6 radians

60 degrees = π/3 radians

90 degrees = π/2 radians

cos 30° = cos(π/6) = a/h =

sin 30° = sin(π/6) = o/h = 1/2

tan 30° = tan(π/6) = o/a =

cos 60° = cos(π/3) = a/h = 1/2

sin 60° = sin(π/3) = o/h =

tan 60° = tan(π/3) = o/a =

Remember 2π radians = 360°. This can be used as a conversion factor when transforming degrees to radians.

• Some basic properties and formulas of the trigonometric functions are:

Sine is an odd function; therefore, for any number Ø

sin (– Ø) = –sin Ø

Cosine is an even function; therefore, for any number Ø

cos (– Ø) = cos Ø

Tangent is an odd function; therefore, for any number Ø

tan (– Ø) = –tan Ø

cot (– Ø) = –cot Ø

sec (– Ø) = sec Ø

csc (– Ø) = –csc Ø

sin (π/2 + Ø) = cos Ø

sin (π/2 – Ø) = cos Ø

cos (π/2 + Ø) = –sin Ø

cos (π/2 – Ø) = sin Ø

sin (90° + Ø) = cos Ø

sin (90° – Ø) = cos Ø

cos (90° + Ø) = –sin Ø

cos (90° – Ø) = sin Ø

tan (90° + Ø) = –cot Ø

tan (90° – Ø)= cot Ø

sin (180° + Ø) = –sin Ø

sin (180° – Ø)= sin Ø

cos (180° + Ø) = –cos Ø

cos (180° – Ø) = –cos Ø

tan (180° + Ø) = tan Ø

tan (180° – Ø) = –tan Ø

sec (90° + Ø) = –csc Ø

sec (90° – Ø) = csc Ø

csc (90° + Ø) = sec Ø

csc (90° – Ø) = sec Ø

cot (90° + Ø) = –tan Ø

cot (90° – Ø) = tan Ø

sec (180° + Ø) = –sec Ø

sec (180° – Ø) = –sec Ø

csc (180° + Ø) = –csc Ø

csc (180° – Ø) = csc Ø

cot (180° + Ø) = cot Ø

cot (180° – Ø) = –cot Ø

sin 2 Ø = 2 sin Ø cos Ø

cos 2 Ø = 2 cos² Ø – 1

tan 2 Ø = 2 tan Ø/1 – tan² Ø

Sum, difference, and product formulas:

cos x – cos y = –2 sin(1/2)(x + y) sin(1/2)(x – y)

cos x + cos y = 2 cos(1/2)(x + y) cos(1/2)(x – y)

sin x – sin y = 2 cos(1/2)(x + y) sin(1/2)(x – y)

sin x + sin y = 2 sin(1/2)(x + y) cos(1/2)(x – y)

sin x cos y = (1/2)sin(x – y) +(1/2)sin(x + y)

cos x sin y = (1/2)sin(x + y) – (1/2)sin(x – y)

cos x cos y = (1/2)cos(x – y) + (1/2)cos(x + y)

sin x sin y = (1/2)cos(x – y) – (1/2)cos(x + y)

• The following identities for two adjacent angles (Ø and Ω) are:

Angles Ø and Ω

sin (Ø + Ω) = sin Ø cos Ω + cos Ø sin Ω

cos (Ø + Ω) = cos Ø cos Ω – sin Ø sin Ω

tan (Ø + Ω) = (tan Ø + tan Ω) / (1 – tan Ø tan Ω)

sin (Ø – Ω) = sin Ø cos Ω – cos Ø sin Ω

cos (Ø – Ω) = cos Ø cos Ω + sin Ø sin Ω

tan (Ø – Ω) = (tan Ø – tan Ω) / (1 + tan Ø tan Ω)

sin Ø cos Ω = (1/2)(sin (Ø + Ω) + sin (Ø – Ω))

cos Ø sin Ω = (1/2)(sin (Ø + Ω) – sin (Ø – Ω))

cos Ø cos Ω = (1/2)(cos (Ø + Ω) + cos (Ø – Ω))

sin Ø sin Ω = (– 1/2)(cos (Ø + Ω) – cos (Ø – Ω))

• Oblique triangles are triangles in planes that are not right triangles. They are described using the Law of Sines, Law of Cosines, and Law of Tangents.

Law of Sines

By rearranging, the following are obtained:

Law of Tangents

• The graphs in this section depict cosine, sine, tangent, secant, cosecant, and cotangent. Cosine, sine, and tangent are described by the following equations:

cos x = cos(x + 2nπ)

cos x = sin(π/2 + x)

sin x = sin(x + 2nπ)

tan x = tan(x + nπ)

Where n is any integer and x is any real number.

It is possible to graph y = cos x and y = sin x by selecting values for x and calculating the corresponding y values.

If there are coefficients in the equations y = cos x and y = sin x, the function will have the same general shape, but it will have a larger or smaller amplitude (taller or shorter), or it will be elongated or narrower, or it will be moved to the right or left or up and down.

For example, if there is a coefficient of 2 in front of cos or sin, the graph will go to + 2 and – 2 (rather than + 1 and – 1) on the y axis.

Similarly, if there is a coefficient of 1/2 in front of cos or sin, the graph will go to + 1/2 and – 1/2 (rather than + 1 and – 1) on the y axis.

If there is a coefficient of 2 in front of x, giving y = cos 2x and y = sin 2x, the graph will complete each cycle along the x axis twice as fast. Because there is one cycle between 0 and 2π for y = cos x and y = sin x, there will be two cycles between 0 and 2π for y = cos 2x and y = sin 2x.

Similarly, if there is a coefficient of 1/2 in front of x, giving y = cos x/2 and y = sin x/2, the graph will complete each cycle along the x axis half as fast. Because there is one cycle between 0 and 2π for y = cos x and y = sin x, there will be one-half of a cycle between 0 and 2π for y = cos 2x and y = sin 2x.

If a number is added or subtracted giving, for example, y = cos x + 2 and y = sin x + 2, the function will be moved up or down on the y axis, in this case, up 2.

• The following graphs depict secant, cosecant, and cotangent:

• Trigonometric functions and exponential functions are related to each other. The following equations define the relationship between these functions.

• Note that . See imaginary numbers in Section 1.17, “Complex Numbers,” in Basic Math and Pre-Algebra, the first book in the Master Math series.

(This is Euler’s identity. It defines the simple relationship between e^iθ, cosθ, and sinθ.)

The expansions for e^x, cos x, and sin x are:

Therefore, the expansions for e^iθ, cos θ, and sin θ are:

• Hyperbolic functions are real and are derived from the exponential functions e^θ and e^–θ. The following equations define hyperbolic functions.

The hyperbolic cosine:

The hyperbolic sine:

The hyperbolic tangent:

The hyperbolic cosecant:

The hyperbolic secant:

The hyperbolic cotangent:

• Note that cosh θ and sinh θ are similar to the functions for cos θ and sin θ