+5 Find the smallest positive solution to
\(\tan 2x + \tan 3x = \sec 3x\)
in radians.
+1763 To solve the equation tan(2x) + tan(3x) = sec(3x), we can first use the identity sec(x) = 1/cos(x) to rewrite the equation as:
tan(2x) + tan(3x) = 1/cos(3x)
Next, we can use the identity tan(x) = sin(x)/cos(x) to rewrite the equation again:
sin(2x)/cos(2x) + sin(3x)/cos(3x) = 1/cos(3x)
Now, we can multiply both sides of the equation by cos(2x)cos(3x) to eliminate the denominators:
sin(2x)cos(3x) + sin(3x)cos(2x) = cos(2x)
Using the double angle and angle addition formulas, we can simplify the left side of the equation:
sin(x)cos(x) = cos(2x)
Finally, we can divide both sides of the equation by cos(x) and use the identity tan(x) = sin(x)/cos(x) to get:
tan(x) = 1/cos(x)
Therefore, the equation is equivalent to tan(x) = sec(x).
To find the smallest positive solution to this equation, we can use the unit circle or a calculator. We can see that the solution is x = π/4.
Therefore, the smallest positive solution to the equation tan(2x) + tan(3x) = sec(3x) is x = π/4 radians.
