Hey guys! Today, we're diving into a fun trigonometric equation: ipsin(1)sesin(5)se = cos(1)secos(5)se. Sounds intimidating? Don't worry, we'll break it down step by step. Our goal is to find the value(s) of 'se' that satisfy this equation. Trigonometric equations can sometimes look complex, but with the right approach and a few key trigonometric identities, we can simplify them and find the solutions. Let's get started!

Initial Assessment

First, let's rewrite the equation to make it a bit clearer. We have:

sin(1) * sin(5se) = cos(1) * cos(5se)

Here, 'se' seems to be the variable we need to solve for. The equation involves products of sine and cosine functions, which suggests we might want to use trigonometric identities to simplify it. One identity that comes to mind is the cosine addition formula:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Our equation looks somewhat similar, but it has a plus sign instead of a minus sign. Let's try to rearrange the equation to see if we can make it fit this form.

Rearranging the Equation

To apply the cosine addition formula, we want to get the terms on one side of the equation. So, let's move everything to the right-hand side:

0 = cos(1)cos(5se) - sin(1)sin(5se)

Now, this looks exactly like the cosine addition formula, where A = 1 and B = 5se. So, we can rewrite the equation as:

cos(1 + 5se) = 0

This is a much simpler equation to solve! We've transformed a complex-looking equation into a basic trigonometric equation.

Solving for 'se'

Now that we have cos(1 + 5se) = 0, we need to find the values of (1 + 5se) for which the cosine function is zero. We know that cosine is zero at odd multiples of π/2. That is:

1 + 5se = (2n + 1) * (π/2), where n is an integer (n = 0, ±1, ±2, ...)

Now, we solve for 'se':

5se = (2n + 1) * (π/2) - 1

se = [(2n + 1) * (π/2) - 1] / 5

So, the general solution for 'se' is:

se = (π(2n + 1)/2 - 1) / 5

Where n is any integer. This means there are infinitely many solutions for 'se', each corresponding to a different integer value of n.

Verifying the Solution

To make sure our solution is correct, let's plug in a few values of n and see if they satisfy the original equation. For example, let's take n = 0:

se = (π/2 - 1) / 5

Now, plug this value of se back into the original equation: sin(1)sin(5se) = cos(1)cos(5se)

sin(1)sin(π/2 - 1) = cos(1)cos(π/2 - 1)

Using the cofunction identities, sin(π/2 - x) = cos(x) and cos(π/2 - x) = sin(x), we get:

sin(1)cos(1) = cos(1)sin(1)

This is true, so our solution is correct for n = 0. You can try other values of n to further verify the solution.

Conclusion

Alright, folks! We've successfully solved the trigonometric equation ipsin(1)sesin(5)se = cos(1)secos(5)se. By using trigonometric identities, specifically the cosine addition formula, we simplified the equation and found the general solution for 'se'. Remember, the key to solving trigonometric equations is to recognize patterns and apply appropriate identities. Keep practicing, and you'll become a pro in no time! The general solution is se = (π(2n + 1)/2 - 1) / 5, where n is any integer. This provides an infinite set of solutions. Good job, everyone!

Extra Practice Problems

To help solidify your understanding, here are a few practice problems related to solving trigonometric equations:

1. Solve sin(2x) = cos(x) for x in the interval [0, 2π].
2. Find the general solution of cos(3θ) = 1/2.
3. Solve 2sin²(x) - sin(x) - 1 = 0 for x in the interval [0, 2π].
4. Determine the general solution for tan(x) + cot(x) = 2.

Tips for Solving Trigonometric Equations

• Know Your Identities: Make sure you are familiar with basic trigonometric identities, such as Pythagorean identities, sum and difference formulas, double angle formulas, and half-angle formulas. These are essential for simplifying and solving trigonometric equations.
• Simplify First: Before attempting to solve an equation, simplify it as much as possible. Look for opportunities to use trigonometric identities to rewrite the equation in a simpler form.
• Isolate Trigonometric Functions: Try to isolate trigonometric functions on one side of the equation. This often makes it easier to apply inverse trigonometric functions or to recognize patterns that lead to a solution.
• Check for Extraneous Solutions: When solving trigonometric equations, especially those involving squaring or taking square roots, check your solutions to make sure they are not extraneous. Extraneous solutions are solutions that satisfy the transformed equation but not the original equation.
• Use the Unit Circle: The unit circle is a valuable tool for visualizing trigonometric functions and their values at different angles. Use it to help you find solutions to trigonometric equations.
• General Solutions: Remember that trigonometric functions are periodic, so trigonometric equations often have infinitely many solutions. When asked to find the general solution, be sure to include all possible solutions by adding integer multiples of the period of the trigonometric function.
• Interval Restrictions: If you are asked to find solutions in a specific interval, make sure to only include solutions that fall within that interval.
• Factoring: If the trigonometric equation can be factored, factor it and then solve each factor separately.
• Substitution: Use substitution to simplify complex equations. For instance, if you have an equation involving sin²(x) and sin(x), you could let y = sin(x) and solve the resulting quadratic equation.
• Graphical Solutions: If you are having trouble solving a trigonometric equation algebraically, consider using a graphing calculator or software to find approximate solutions graphically.

Common Mistakes to Avoid

• Forgetting the ± Sign: When taking the square root of both sides of an equation, remember to include both the positive and negative square roots.
• Dividing by Zero: Be careful not to divide by zero when solving trigonometric equations. For example, if you have an equation of the form sin(x)cos(x) = cos(x), don't divide both sides by cos(x) without considering the case where cos(x) = 0.
• Incorrectly Applying Identities: Make sure you are using trigonometric identities correctly. Double-check the formulas and be careful with signs.
• Ignoring the Periodicity: When finding general solutions, make sure to account for the periodicity of the trigonometric functions. For example, the general solution to sin(x) = 0 is x = nπ, where n is an integer.
• Not Checking Solutions: Always check your solutions to make sure they satisfy the original equation and are not extraneous.
• Mixing Degrees and Radians: Be consistent with your units. If the equation is given in degrees, make sure to work in degrees. If it's given in radians, work in radians. If necessary, convert between degrees and radians using the conversion factor π radians = 180 degrees.