Solve the Trig Equation sin(x/2) = cos(x/2) for the values of x

TL;DR

Learn how to solve a trigonometric equation step-by-step using manipulation and substitution.

Transcript

in this problem we have a trigonometric equation and we're being asked to solve for x so whenever you have a trig equation and it's not just like sine x or cosine x whenever it's something besides x inside the trig function the approach is the following you start by writing this down so we have that x is less than two pi and is greater than or equa... Read More

Key Insights

• 💠 Trigonometric equations with variables inside the trig function can be solved by manipulating the equation to resemble what's inside.
• 🍉 Dividing each term by the coefficient of the variable isolates the variable and simplifies the equation.
• 💄 By renaming the variable, such as 'u,' the equation can be transformed, making it easier to solve.
• 🤨 The values that satisfy the equation sine(u) = cosine(u) lie between 0 and pi on the unit circle.
• 🔺 The angles pi/4 and 5pi/4 are the only angles where sine and cosine are equal.
• ☺️ Substituting back, the solution to the original equation is x = pi/2.
• 🤩 Manipulation and substitution are the key steps in solving trigonometric equations.

Questions & Answers

Q: What is the approach to solving a trigonometric equation with a variable inside the trig function?

The approach involves manipulating the equation to make it resemble the expression inside the trig function. Divide each term by the coefficient to isolate the variable and simplify the equation.

Q: How can the equation be transformed to make it easier to solve?

By renaming the variable with a convenient letter, such as 'u,' the equation can be rewritten as sine(u) = cosine(u). This simplification makes it easier to find the solutions.

Q: What values of 'u' satisfy the equation sine(u) = cosine(u)?

The equation is satisfied when 'u' lies between 0 and pi on the unit circle. This includes the angle pi/4 and 5pi/4, where sine and cosine are equal to sqrt(2)/2 (positive values) and -sqrt(2)/2 (negative values).

Q: What is the solution to the original equation x = 2u?

Substituting back, the solution to the equation x = 2u is x = pi/2. Multiplying both sides of 2u = pi/4 by 2 gives the result.

Summary & Key Takeaways

• To solve a trigonometric equation with a variable inside the trig function, manipulate the equation to make it look like what's inside the function.

• Divide each piece by the coefficient to isolate the variable and rename it for convenience.

• Find the values where the trig functions are equal by considering the unit circle.

• Substituting back, the solution to the equation is x = pi/2.