Solving Trigonometric Equations

A trigonometric equation is an equation that involves trigonometric functions of a variable, typically an angle. These equations can be solved for the variable to find the angle(s) that satisfy the equation.

1. Isolate the trigonometric function. 2. Use trigonometric identities if necessary. 3. Solve for the angle(s). 4. Consider the periodic nature of trigonometric functions to find all solutions within the specified interval.

The solution set is $$\text{2\text{pi}, } \frac{7\text{pi}}{6}, \frac{11\text{pi}}{6}$$ .

Rearrange to form a quadratic equation in terms of $$\text{sin}x$$ : $$50\text{sin}^2x - 75\text{sin}x - 50 = 0$$ . Use the quadratic formula to find $$\text{sin}x$$ , then solve for $$x$$ .

The solutions are $$\frac{7\text{pi}}{6}, \frac{11\text{pi}}{6}$$ .

The interval specifies the range of values for the variable (usually an angle) within which solutions must be found. It helps to limit the number of solutions to those that are relevant.

This equation can be factored to find solutions for $$\text{sin}\, x = 0$$ or $$\tan x = -1$$ .

The solutions are $$\frac{3\text{pi}}{4}, \text{pi}, \frac{7\text{pi}}{4}, 2\text{pi}$$ .

The sine function has a periodicity of $$2\text{pi}$$ , meaning it repeats its values every $$2\text{pi}$$ radians.

The cosine function also has a periodicity of $$2\text{pi}$$ , repeating its values every $$2\text{pi}$$ radians.