Find the solution to the follow trig equations on the interval [0,2pi). Round to the nearest ten thousandth. If there are multiple values, enter your answers from least to greatest separated by commas without spaces. Example: .1234,.5678,.9101
1) tan(3x)=1
2) 3sec^2x=4
tan(3x)=1
Let p=3x
Tanp=1
P=tan-¹(1)
P=π/4=45°
Where
P=3x
Tan has a period of π
3x=π/4+πn
X=π/12+πn/3
Where n represent integer
Similar
3sec^2x=4
3/cos²x=4
Cos²x=3/4
Cosx=√3/2=30° or π/6
Cos has a period movement of 2π
x=π/6+2nπ
What ever you do make sure you don't exceed 2π
If you fancy degree (π=180)
I find these are easiest done this way:
tan(3x)=1
I know that tan 45° =1 or tan225° = 1, (using the CAST rule)
3x = 45° or 3x = 225°
x = 15° or x = 75°
but the period of tan(3x) is 360/3 = 120°
so to find a new solution all we have to do is add 120° to any current solution
to stay within the 0 ≤ x ≤ 2π domain
x = 15°, 135°, 255°, 75°, 195°, or 315°
or in radians, π/12, 9π/12, 17π/12, 13π/12, 21π/12
3sec^2x=4
sec^2x=4/3
secx = ± 2/√3 , so x can be in all 4 quadrants
cosx = √3/2 , recognize x = 30°
x = 30°, 150°, 210°, or 330°
or the equivalent radians of π/6, 5π/6, 7π/6 and 11π/6
Cool I like the way you think smart .......you make maths fun to learn here
To find the solutions to the trigonometric equations, we will apply the following steps for each equation:
1) tan(3x) = 1
Step 1: Determine the general form of the equation.
Since we are looking for the values of x that satisfy the equation, we need to isolate x.
tan(3x) = 1
Step 2: Use inverse trigonometric functions to find the values of x.
We can use the inverse tangent (arctan) function to find the values of x.
arctan(tan(3x)) = arctan(1)
3x = arctan(1)
Step 3: Simplify the equation and solve for x.
Using the trigonometric identity tan(arctan(x)) = x, we can simplify the equation.
3x = π/4 + πn, where n is an integer
x = (π/4 + πn)/3
To find the values of x in the interval [0,2π), we substitute the values of n that will give us the solutions within this interval.
When n = 0, we have:
x = (π/4)/3 = π/12
When n = 1, we have:
x = (π/4 + π)/3 = 5π/12
Since n = 2π will result in the same values as n = 0 or 1, we can stop here.
The solutions to the equation tan(3x) = 1 on the interval [0,2π) are approximately:
x ≈ π/12, 5π/12
Rounding to the nearest ten thousandth, the solutions are approximately:
x ≈ 0.2618, 1.0472
2) 3sec^2(x) = 4
Step 1: Determine the general form of the equation.
Since we are looking for the values of x that satisfy the equation, we need to isolate x.
3sec^2(x) = 4
Step 2: Use inverse trigonometric functions to find the values of x.
We can use the inverse secant (arcsec) function to find the values of x.
sec^2(x) = 4/3
Step 3: Simplify the equation and solve for x.
Taking the square root of both sides of the equation:
sec(x) = ±√(4/3)
Using the definition of the secant function as the reciprocal of the cosine function:
cos(x) = ±√(3/4)
Since we are given the interval [0,2π), we can determine the values of x by considering the cosine function in quadrants I and II where it is positive.
In quadrant I:
cos(x) = √(3/4)
x = arcsin(√(3/4)), where 0 ≤ x < π/2
In quadrant II:
cos(x) = -√(3/4)
x = π - arcsin(√(3/4)), where π/2 < x ≤ π
Round the solutions to the nearest ten thousandth.
The solutions to the equation 3sec^2(x) = 4 on the interval [0,2π) are approximately:
x ≈ 0.9016, 2.2392