tan(3x) + 1 = sec(3x)
Thanks,
pretend 3x equals x
so tanx + 1 = secx
we know the law that 1 + tanx = secx
so tanx + 1 becomes secx
and... secx = secx
sec(3x) = sec(3x) [just put 3x back in for x- you don't really have to change 3x to x but it kinda makes it easier
Let 3x =u for simplicity sake
then tan u + 1 = sec u
sinu/cosu + 1 = 1/cosu
multiply by cos u
sin u + cos u = 1
square both sides
sin^2 u + 2(sinu)(cosu) + cos^2 u =1
but sin^A+cos^2Ax=1
so 1 + 2(sinu)(cos)=1
recall that sin(2A_=2sinAcosS
then sin 2u =0
or replacing the u
sin 6x = 0
by the CAST rule, 6x = 0,180º,360º
or 0, pi/2, pi in radians
so x = 0, 30º,60º or 0,pi/6,pi/3
The period of tan(3x)=180º or pi radians
so other answers can be obtained by adding mulitples of 180º or multiples of pi radians to any of the above answers.
eg. take the 60º answer, if we add 5*180 to it we get 960º
Left Side: tan (3*960) +1 = 0 + 1 = 1
Right Side: sec(3*960)= 1
sorry haley, that is wrong
tanx + 1 is not equal to secx
rather
tan^2 x + 1 = sec^2 x
see my solution below.
I forgot to include the following:
Since "squaring" took place in my solution, all answers should have been verfied.
Upon checking, we find that 30º does not work, since tan90º is undefined.
So all periodic answers based on 30º do not work
sin (u) + cos (u) = 1 --->
sqrt(2)sin(u+pi/4) = 1
Note that
sin(a+b) = sin(a)cos(b) + cos(a) sin(b)
You can use this rule to write a sum of sin and cos as a single sin or cos.
sin(u+pi/4) = 1/sqrt[2] --->
u = 0 Mod(2 pi) OR u = pi/2 Mod(2 pi)
In this case you could have found the two solutions u = 0 and
u = pi/2 by inspection. Because there can be only two solutions in an interval of 2 pi, you then know that these are all the solutions in such an interval. All other solutions differ from these by a multiple of 2 pi.
Wow, that's quite a lengthy explanation! It looks like you did all the math correctly, but let me try to lighten the mood a bit with a joke:
Why did the sine wave bring a flashlight to the party?
Because it wanted to find its cosine! (co-sign) 😄
To solve the equation tan(3x) + 1 = sec(3x), we can simplify it as follows:
1. Let's substitute 3x with u for simplicity, so the equation becomes:
tan(u) + 1 = sec(u)
2. We know that the identity 1 + tan(u) = sec(u) holds true, so the equation can be rewritten as:
sec(u) = sec(u)
3. Now, let's focus on solving sec(u) = sec(u).
Recall that sec(u) = 1/cos(u) and sec(u) is defined for all real numbers except where cos(u) is equal to zero.
4. Multiply both sides of the equation by cos(u) to eliminate the denominator:
sin(u) + cos(u) = 1
5. Square both sides of the equation:
sin^2(u) + 2sin(u)cos(u) + cos^2(u) = 1
6. Use the identity sin^2(u) + cos^2(u) = 1 to simplify the equation:
1 + 2sin(u)cos(u) = 1
7. Simplify 2sin(u)cos(u) to sin(2u) using the double-angle formula for sine:
1 + sin(2u) = 1
8. Solve for sin(2u) = 0:
sin(2u) = 0
9. Find the values of u that satisfy sin(2u) = 0.
Using the periodicity of sine, we can find the solutions for u:
u = 0, pi/2, pi, 3pi/2, 2pi, 5pi/2, ...
10. Substituting back 3x for u, we get:
3x = 0, pi/2, pi, 3pi/2, 2pi, 5pi/2, ...
11. Simplify the values of x:
x = 0, pi/6, pi/3, pi/2, 2pi/3, 5pi/6, ...
Note: Because tan(3x) and sec(3x) are periodic functions, there are infinitely many solutions. The values listed are for one full period of the functions. To obtain additional solutions, you can add multiples of the period (180 degrees or pi radians) to any of the above answers.
To solve the equation tan(3x) + 1 = sec(3x), we can use the trigonometric identity tan^2 x + 1 = sec^2 x.
1. First, let's rewrite the equation as tan^2(3x) + 1 = sec^2(3x).
2. Now, substitute u = 3x to simplify the equation. It becomes tan^2(u) + 1 = sec^2(u).
3. Using the trigonometric identity tan^2 x + 1 = sec^2 x, we can rewrite the equation as sec^2(u) = sec^2(u).
4. Simplifying sec^2(u) = sec^2(u), we get the identity true for all values of u.
5. Since 3x = u, we can conclude that sec(3x) = sec(u) is true for all values of x.
From here, we need to find the values for which tan(3x) + 1 = sec(3x), or equivalently, tan(3x) + 1 = tan(3x).
To solve this, we can use the trigonometric identity sin^2 u + cos^2 u = 1.
1. From the equation tan(3x) + 1 = tan(3x), we can subtract tan(3x) from both sides to get 1 = 0.
2. This equation is not possible, so there are no solutions to tan(3x) + 1 = sec(3x).
Therefore, the original equation tan(3x) + 1 = sec(3x) has no solutions.