Sin²(x)=0.25
actually, there are 4 solutions: 30°, 150°, 210°, 330°
You are correct. Since the sine function is periodic with a period of 360 degrees (or 2π radians), there are actually four solutions to the equation sin²(x) = 0.25 within one period:
1) x = 30 degrees or π/6 radians
Taking the sine of 30 degrees, we get sin(30) = 0.5, so
sin²(30) = (0.5)² = 0.25
2) x = 150 degrees or 5π/6 radians
Taking the sine of 150 degrees, we get sin(150) = -0.5 (since sine is negative in the second quadrant), so
sin²(150) = (-0.5)² = 0.25
3) x = 210 degrees or 7π/6 radians
Taking the sine of 210 degrees, we get sin(210) = -0.5 (since sine is negative in the third quadrant), so
sin²(210) = (-0.5)² = 0.25
4) x = 330 degrees or 11π/6 radians
Taking the sine of 330 degrees, we get sin(330) = 0.5 (since sine is positive in the fourth quadrant), so
sin²(330) = (0.5)² = 0.25
So, there are four solutions to the equation sin²(x) = 0.25 within one period.
To solve the equation sin²(x) = 0.25, we need to find the values of x that satisfy this equation. Here's how you can do it step by step:
Step 1: Take the square root of both sides of the equation to eliminate the square. Remember to consider both the positive and negative square root:
√(sin²(x)) = ±√(0.25)
Step 2: Simplify the equation:
sin(x) = ±0.5
Step 3: Find the angles where sin(x) equals ±0.5. Recall the values of sine for some key angles; sin(30°) = 0.5 and sin(150°) = 0.5. Similarly, sin(210°) = -0.5 and sin(330°) = -0.5.
Therefore, the possible values of x are:
x = 30° + 360°n, where n is an integer
x = 150° + 360°n, where n is an integer
x = 210° + 360°n, where n is an integer
x = 330° + 360°n, where n is an integer
These are the values of x that satisfy the equation sin²(x) = 0.25.
There are two possible solutions to this equation:
1) x = 30 degrees or pi/6 radians
Taking the sine of 30 degrees, we get sin(30) = 0.5, so
sin²(30) = (0.5)² = 0.25
2) x = 150 degrees or 5pi/6 radians
Taking the sine of 150 degrees, we get sin(150) = -0.5 (since sine is negative in the second quadrant), so
sin²(150) = (-0.5)² = 0.25