Question

Solve sec(3x) - 6 = 0 for the four smallest positive solutions.

Answer 1

To solve the equation sec(3x) - 6 = 0, follow these steps: add 6 to both sides, take the reciprocal, take the arccosine, divide by 3, and find the four smallest positive solutions.

Step-by-step explanation:

Solving the equation sec(3x) - 6 = 0:

Step 1: Add 6 to both sides of the equation: sec(3x) = 6.

Step 2: Take the reciprocal of both sides to get cos(3x) = 1/6.

Step 3: Take the arccosine of both sides to find the angle: 3x = arccos(1/6).

Step 4: Divide both sides by 3: x = arccos(1/6)/3.

Step 5: Repeat step 4 with additional multiples of 2π to find the four smallest positive solutions.

Answer 2

The four smallest positive solutions to the trigonometry function are 26.80, 28.90, 30.99, 33.09

How to solve trigonometric equations.

The solution to the trigonometric equation sec(3x) -6 = 0 can be determined by using the following process.

Given that:

sec (3x) - 6 = 0

Add 6 to both sides;

sec (3x) = 6

Using trigonometric rule;

`cos (3x) = (1)/(6)`

`3x = cos^(-1) ((1)/(6))`

`x = (1)/(3)cos^(-1) ((1)/(6))`

Now, the standard general solution for the equation is:

`x = (1)/(3)cos^(-1) ((1)/(6)) + k((2\pi)/(3))`

• here, k represents the integer 0, 1, 2, 3.

To find the four smallest positive solutions for k's

`x|_(k=0)=(1)/(3) cos^(-1)((1)/(6))+0=26.80`

`x|_(k=1)= (1)/(3) cos^(-1)((1)/(6))+ ((2\pi)/(3))=28.90`

`x|_(k=2)= (1)/(3) cos^(-1)((1)/(6)) +((4\pi)/(3))=30.99`

`x|_(k=3)= (1)/(3) cos^(-1)((1)/(6)) +((6\pi)/(3))=33.09`