( 0,\ \pi/3,\ \pi,\ 5\pi/3 ). Type 4: Equation with tangent Example: ( \tan(2x) = 1 ) in ( [0, 2\pi) ).
Let ( t = 2x ). Solve ( \tan t = 1 ). Principal value: ( t = \pi/4 ). Tangent period is ( \pi ): ( t = \pi/4 + k\pi ). Thus ( 2x = \pi/4 + k\pi \Rightarrow x = \pi/8 + k\pi/2 ).
Method 1: Divide by ( \cos x ) (if ( \cos x \neq 0 )): ( \tan x = 1 \Rightarrow x = \pi/4 + k\pi ). Check ( \cos x = 0 ) gives ( x = \pi/2, 3\pi/2 ), but those don’t satisfy original (sine ≠ cosine). So fine. ecuaciones trigonometricas 1 bachillerato
With practice, solving trigonometric equations becomes systematic. Memorize the general solution forms and always check your solutions in the original equation.
Find ( k ) for ( 0 \le x < 2\pi ): ( k=0 \to \pi/8 ) ( k=1 \to \pi/8 + \pi/2 = 5\pi/8 ) ( k=2 \to 9\pi/8 ) ( k=3 \to 13\pi/8 ) ( k=4 \to 17\pi/8 = 2\pi + \pi/8 ) (too large). ( 0,\ \pi/3,\ \pi,\ 5\pi/3 )
Bring all terms: ( \sin x \cos x - \frac12\sin x = 0 ) ( \sin x (\cos x - 1/2) = 0 )
1. What is a Trigonometric Equation? A trigonometric equation is an equation where the unknown variable appears as the argument of one or more trigonometric functions (sine, cosine, tangent, etc.). Solve ( \tan t = 1 )
( \pi/8,\ 5\pi/8,\ 9\pi/8,\ 13\pi/8 ). Type 5: Equation with sine and cosine of the same angle Example: ( \sin x = \cos x ).