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Trigonometric Functions for Physics Help

By — McGraw-Hill Professional
Updated on Sep 17, 2011

Introduction

There are six basic trigonometric functions . They operate on angles to yield real numbers and are known as sine, cosine, tangent, cosecant, secant , and cotangent . In formulas and equations, they are abbreviated sin, cos, tan, cos, sec , and cot , respectively.

Until now, angles have been denoted using lowercase italicized English letters from near the end of the alphabet, for example, w, x, y , and z . In trigonometry, however, Greek letters are almost always used, particularly θ (italicized lowercase theta , pronounced “THAY-tuh”) and ϕ (italicized lowercase phi , pronounced “FIE” or “FEE”). We will follow this convention here. You should get used to it so that you know how to pronounce the names of the symbols when you see them. This will help you avoid embarrassment when you’re around physicists. More important, having a pronunciation in your “mind’s ear” may make it easier for you to work with formulas containing such symbols.

Basic Circular Functions

Consider a circle in rectangular coordinates with the following equation:

x 2 + y 2 = 1

This is called the unit circle because its radius is 1 unit, and it is centered at the origin (0, 0), as shown in Fig. 5-9. Let θ be an angle whose apex is at the origin and that is measured counterclockwise from the abscissa ( x axis). Suppose that this angle corresponds to a ray that intersects the unit circle at some point P = ( x 0 , y 0 ). Then

y 0 = sin θ

x 0 = cos θ

y 0 / x 0 = tan θ

Logarithms, Exponentials, and Trigonometry Trigonometric Functions Basic Circular Functions

Fig. 5-9 Unit-circle model for defining trigonometric functions.

Secondary Circular Functions

Three more circular trigonometric functions are derived from those just defined. They are the cosecant function, the secant function, and the cotangent function. In formulas and equations, they are abbreviated csc θ, sec θ , and cot θ . They are defined as follows:

cos θ = 1/(sin θ) = 1/ y 0

sec θ = 1/(cos θ) = 1/ x 0

cot θ = 1/(tan θ) = x 0 / y 0

Right-triangle Model

Consider a right triangle Δ PQR such that ∠ PQR is the right angle. Let d be the length of line segment RQ, e be the length of line segment QP , and f be the length of line segment RP , as shown in Fig. 5-10. Let θ be the angle between line segments RQ and RP . The six circular trigonometric functions can be defined as ratios between the lengths of the sides as follows:

sin θ = e / f

cos θ = d / f

tan θ = e / d

cos θ = f / e

sec θ = f / d

cot θ = d / e

Logarithms, Exponentials, and Trigonometry Trigonometric Functions Right-triangle Model

Fig. 5-10 Right-triangle model for defining trigonometric functions.

 

 Practice problems for these concepts can be found at:  Logarithms, Exponentials, And Trigonometry for Physics Practice Test

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