How do u solve equations like
sin 2x= tan 9x+3
Do you mean (tan9x) +3 or tan (9x+3) ?
Wow, that is a nasty one.
Did you make it up or is that actually a question from a textbook?
I started breaking tan(9x) down into
tan(x+8x) and then started using the half-angle formulas to get tan 4x , then tan 2x, but the back-substitution became horrendous.
I sometimes use this very powerful
equation solver
http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=equations&s2=solve&s3=basic
It gave me about 18 real solutions for x, of course they would be all in radians
I used the interpretation
sin(2x) = tan(9x) + 3
I tested the first answer of x= 1.60579 and another x = 1.26517 and they both worked.
try it.
The answer depends upon your answer to my first question. It is a messy problem either way
To solve the equation sin 2x = tan 9x + 3, we will first simplify the equation using trigonometric identities. Then, we will solve for the unknown variable, x, using algebraic techniques. Let's go step by step:
Step 1: Simplify the equation
Start by using a trigonometric identity to rewrite the equation in terms of a single trigonometric function. Recall the following identities:
- sin 2x = 2sin x cos x
- tan x = sin x / cos x
Using these, we can rewrite the equation as:
2sin x cos x = (sin 9x + 3cos 9x) / cos 9x
Step 2: Eliminate the fraction
To eliminate the fraction in the equation, we can multiply both sides of the equation by cos 9x. This will give us:
2sin x cos x cos 9x = sin 9x + 3cos 9x
Step 3: Expand the equation
Using the double angle formula for sine (sin 2θ = 2sin θ cos θ), we can expand the left side of the equation to get:
sin 9x = sin 9x + 3cos 9x
Step 4: Simplify the equation
Now, we can simplify the equation by canceling out like terms:
0 = 3cos 9x
Step 5: Solve for x
Since cos 9x cannot be zero (because the cosine function oscillates between -1 and 1), the equation 0 = 3cos 9x has no solutions.
Therefore, the equation sin 2x = tan 9x + 3 has no solutions for x.