**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

**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*

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

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