5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:46

Review of Precalculus 65

0

y = cot x

x

y

− 2 π−1. 5π − 1 π−0.5π 0.5π 1 π 1. 5π 2 π

Domain: {all x ≠ 0, ±π, ± 2 π}

Range: {−∞ < y < ∞}

Frequency: 1 Period: π

Figure 5.3-19

0

y = cot−^1 x

x

y

− 3 − 2 − 1123

π

Domain: {−∞ < x < ∞}

Range: {0 < y < π}

Figure 5.3-20

Formulas for using a calculator to get sec−^1 x, csc−^1 x, and cot−^1 x:

sec−^1 x=cos−^1 (1/x)

csc−^1 x=sin−^1 (1/x)

cot−^1 x=π/ 2 −tan−^1 x

Example 1

Sketch the graph of the functiony=3 sin 2x. Indicate its domain, range, amplitude, period,

and frequency.

The domain is all real numbers. The range is [−3, 3]. The amplitude is 3, which is the

coefficient of sin 2x. The frequency is 2, the coefficient ofx, and the period is (2π)÷(the

frequency), thus 2π÷ 2 =π. (See Figure 5.3-21.)

− 2

− 3

− 1

0

y = 3 sin (2x)

x

y

− 2 π−1. 5π − 1 π−0.5π 0.5π 1 π 1. 5π 2 π

3

1

2

Figure 5.3-21

Example 2

Solve the equation cosx=− 0 .5if0≤x≤ 2 π.

Note that cos (π/3)= 0 .5 and that the cosine is negative in the second and third quadrants.

Since cosx=− 0 .5,xmust be in the second or third quadrant with a reference angle ofπ/3.

In the second quadrant,x=π−(π/3)= 2 π/3 and in the third quadrant,x=π+(π/3)= 4 π/3.

Thusx= 2 π/3or4π/3. (See Figure 5.3-22.)