(3 tan^2x-1)(tan^2-3)= 0
can someone please explain how to do this problem and then explain the whole pie thing at the end...like, this is suppose to equal pie/6 b/c it's tan...im soooo confused.....
(3 tan^2x-1)(tan^2-3)= 0
so 3tan^2x = 1 or tan^2x = 3
tan^2x = 1/3 or tan^2x = -3
the second part has no solution, since something^2 cannot be negative
tanx = ± 1/√3
so x is in any of the 4 quadrants.
form the 30-60-90 triangle we know that
tan 30°
so in degrees, x = 30°, 150°, 210° and 330°
It appears that we are to give the answers in radians, and you seem to know very little about radians.
I am sure your texbook has a unit on it, or else you can learn about it on the thousands of webpages found after googling "radians"
in short, πradians = 180°
so π/6 = 30° , by dividing both sides of the equation above by 6
further:
150° = 5π/6
210° = 7π/6
330° = 11π/6
Your equation is already factored. You have a solution if either factor is zero.
tan^2x can be either 1/3 or 3
tan x = +or- 1/sqrt3
tanx = +or- sqrt3
x = 30, 60, 120, 150, 210, 240, 300 or 330 degrees
pi/6 radians (30 degrees) is only one of the answers.
Go with drwls solution.
somehow my +3 changed into a -3 for the second factor. there are 4 more solutions from that, as drwls shows.
To solve the equation (3tan^2x-1)(tan^2-3)=0 and understand the connection to pi/6, we need to follow a step-by-step process.
Step 1: Factoring
First, we notice that the equation is a product of two factors: (3tan^2x-1) and (tan^2-3). To find the solutions, we set each factor equal to zero and solve for x separately.
Setting 3tan^2x - 1 = 0:
Add 1 to both sides:
3tan^2x = 1
Divide both sides by 3:
tan^2x = 1/3
Setting tan^2 - 3 = 0:
Add 3 to both sides:
tan^2 = 3
Step 2: Solving for tan(x)
To solve for tan(x), we need to take the square root of both sides of each equation. However, we need to be careful when dealing with trigonometric functions since they have periodicity. To simplify the process, we can use the unit circle and the knowledge of the trigonometric ratios for the special angles.
Let's solve tan^2x = 1/3:
Taking the square root of both sides, we get:
tan(x) = ±√(1/3)
Since tangent is positive in the first and third quadrants, we have two possible solutions:
1) tan(x) = √(1/3)
2) tan(x) = -√(1/3)
To solve tan^2x = 3:
Taking the square root of both sides, we get:
tan(x) = ±√3
Similarly, since tangent is positive in the first and third quadrants, we have two more possible solutions:
3) tan(x) = √3
4) tan(x) = -√3
Step 3: Determining the values of x in radians
Now, we need to determine the actual values of x in radians corresponding to each solution. This is where the concept of the unit circle comes into play.
For the first solution tan(x) = √(1/3), we find the angle by looking for a reference angle that corresponds to the given value of tangent. In this case, the reference angle is π/6 radians (30 degrees) because tan(π/6) = √(1/3).
Thus, the first solution is x = π/6.
Similarly, for the other three solutions:
- tan(x) = -√(1/3) corresponds to x = -π/6
- tan(x) = √3 corresponds to x = π/3
- tan(x) = -√3 corresponds to x = -π/3
Therefore, the complete solution to the equation (3tan^2x-1)(tan^2-3)=0 is:
x = π/6, -π/6, π/3, or -π/3.
In summary, solving trigonometric equations involves factoring, finding solutions to each factor, considering the periodicity of trigonometric functions, and using the unit circle to determine specific angle values in radians.