## Calculus AB and Calculus BC

## CHAPTER 1 Functions

### D. TRIGONOMETRIC FUNCTIONS

The fundamental trigonometric identities, graphs, and reduction formulas are given in the Appendix.

**D1. Periodicity and Amplitude.**

The trigonometric functions are periodic. A function *f* is *periodic* if there is a positive number *p* such that *f* (*x* + *p*) = *f* (*x*) for each *x* in the domain of *f*. The smallest such *p* is called the *period* of *f*. The graph of *f* repeats every *p* units along the *x*-axis. The functions sin *x*, cos *x*, csc *x*, and sec *x*have period 2π; tan *x* and cot *x* have period π.

The function *f* (*x*) = *A* sin *bx* has amplitude *A* and period *g*(*x*) = tan *cx* has period

**EXAMPLE 10**

Consider the function *f* (*x*) = cos (*kx*).

(a) For what value of *k* does *f* have period 2?

(b) What is the amplitude of *f* for this *k* ?

**SOLUTIONS:**

**(a)** Function *f* has period since this must equal 2, we solve the equation getting *k* = π.

**(b)** It follows that the amplitude of *f* that equals has a value of

**EXAMPLE 11**

Consider the function

Find (a) the period and (b) the maximum value of *f*.

(c) What is the smallest positive *x* for which *f* is a maximum?

(d) Sketch the graph.

**SOLUTIONS:**

**(a)** The period of *f* is or 6.

**(b)** Since the maximum value of −sin *x* is −(−1) or +1, the maximum value of *f* is 3 + 1 or 4.

**(c)** equals +1 when that is, when Solving yields

**(d)** We graph in [−5,8] × [0,5]:

**FIGURE N1–6**

**D2. Inverses.**

**Inverse trig functions**

We obtain *inverses* of the trigonometric functions by limiting the domains of the latter so each trigonometric function is one-to-one over its restricted domain. For example, we restrict

The graphs of *f* (*x*) = sin *x* on and of its inverse *f* ^{−1}(*x*) = sin^{−1} *x* are shown in Figure N1–7. The inverse trigonometric function sin^{−1} *x* is also commonly denoted by arcsin *x*, which denotes *the* angle whose sine is *x*. The graph of sin^{−1} *x* is, of course, the reflection of the graph of sin *x* in the line *y* = *x*.

**FIGURE N1–7**

Also, for other inverse trigonometric functions,

*y* = cos^{−1} *x* (or arccos *x*) has domain −1 *x* 1 and range 0 *y* π;

*y* = tan^{−1} *x* (or arctan *x*) has domain the set of reals and range

Note also that