Solving Trigonometric Equations: Techniques and Insights

Solving the Equation cosx sinx

It is quite obvious that if cosx sinx there is a solution. This can be found by equating the two expressions and solving for x. One straightforward solution is:

x π/4 or 5π/4.

From a quick sketch of the functions sinx and cosx, it is evident that these functions intersect at x π/4 and x 5π/4. By symmetry and the periodicity of sine and cosine functions, these are the only solutions modulo 2π.

Another Approach

Instead of directly equating the functions, we can solve the equation:

sinx cosx.

By factoring, we get:

x π/4 (mod π).

A simple plot confirms that these are all the solutions in ?. The only remaining step is to rigorously show that there are no other solutions in the interval [0, 2π]. This can be accomplished through the application of derivative tests or other calculus techniques.

Exploring Another Equation: 3^{sinx - cosx} tanx

To solve the equation 3^{sinx - cosx} tanx, let's begin by drawing the graphs of both sides of the equation.

By inspection, we can see that when x 45° or x 225°, the left-hand side of the equation (LHS) and the right-hand side of the equation (RHS) are both equal to 1. This is because:

LHS 3^{sinx - cosx} 3^{0} 1, and

RHS tanx tan45° 1.

Therefore, we conclude that the equation is satisfied when x 45° or x 225°.

A More Abstract Approach

To solve this in a more abstract manner, let u sinx and v cosx. Substituting these into the equation, we get:

u3^v - v3^u 0.

Rearranging the terms, we obtain:

u3^v v3^u.

Notice that the left-hand side is the same as the right-hand side with the variables transposed. This suggests that if u v, then we have a solution. Therefore:

sinx cosx.

As a result, the solutions are:

x π/4 nπ

where n is an integer. However, to prove that this is the only solution can be quite messy and might require calculus. If you are interested in the detailed proof based on calculus, please let me know.

Conclusion

In conclusion, solving trigonometric equations can be approached in various ways such as graphical methods, substitution, and algebraic manipulation. Each method provides its unique insights and can be adapted to different types of equations. Whether you are looking for quick solutions or detailed proofs, understanding these techniques is key to mastering trigonometry.