Answer to Question #311455 in Trigonometry for bookaddict

Question

The function 𝑓(𝑥) is defined as 𝑓(𝑥)=𝑎+𝑏cos𝑥, where 𝑎 and 𝑏 are constants. The range of 𝑓(𝑥) is given by −6≤𝑓(𝑥)≤2.

a. Find the values of 𝑎 and 𝑏

b. Solve the equation 𝑓(𝑥)=0 for 0°≤𝑥≤360°

c. Sketch the graph of 𝑦=𝑓(𝑥) for 0°≤𝑥≤360°

Answer 1

a)

Because the min value of the function = -6: $a+bcosx =-6$

max value = 2: $a+bcosx = 2$

Because $-1≤cosx≤1$ it means $min(cosx)=-1, max(cosx)=1$

Make up a system:$\begin{cases} a+b(-1)=-6 \\a+b*1=2 \end{cases}$ $\implies$ $\begin{cases} a-b=-6 \\ a+b=2 \end{cases}$ $\implies$ $\begin{cases} 2a=-4 \\ a+b = 2 \end{cases}$ $\implies\begin{cases} a =-2 \\ b=4 \end{cases}$

Answer:$f(x)=-2+4cosx$

b)

$-2+4cosx=0, x\isin[0;2\pi]$

$4cosx=2\implies cosx=\frac24 \implies cosx = \frac 12$

$x=\begin{smallmatrix}+\\- \end{smallmatrix}$ $\frac\pi3+2\pi n, n\isin Z, Z =$ set of integer

for $n=0:$ $x =\frac\pi3$ and $x=-\frac\pi3$ - not suitable

for $n =1:x=\frac\pi3+2\pi$ - not suitable, and $x=-\frac\pi3 +2\pi = \frac{5\pi}3$

for $n=-1: x= \frac\pi3-2\pi$ - not suitable, and $x=-\frac\pi3-2\pi$ - not suitable

Answer: $x=\frac\pi3, x=\frac{5\pi}3$

c)

The required plot of the graph between the blue and green line in the screenshot