given f(x)=sin2x and g(x)=cos(x+30)
1. solve for x if sin2x=cos(x+30) for x€[-180;180]
To solve the equation sin2x = cos(x+30), we need to find the values of x that satisfy this equation within the given range x€[-180;180]. Here's how you can do it step by step:
Step 1: Apply the trigonometric identity: sin(2x) = 2sin(x)cos(x). Rewrite the equation as 2sin(x)cos(x) = cos(x+30).
Step 2: Rearrange the equation to bring all terms to one side: 2sin(x)cos(x) - cos(x+30) = 0.
Step 3: Simplify the equation further by using the trigonometric identity: cos(x+30) = cos(x)cos(30) - sin(x)sin(30). The equation becomes 2sin(x)cos(x) - cos(x)cos(30) + sin(x)sin(30) = 0.
Step 4: Combine like terms: sin(x)(2cos(x) + sin(30)) - cos(x)cos(30) = 0.
Step 5: Recall that sin(30) = 0.5 and cos(30) = √3/2. Substitute these values into the equation: sin(x)(2cos(x) + 0.5) - cos(x)(√3/2) = 0.
Step 6: Further simplify the equation: 2sin(x)cos(x) + 0.5sin(x) - (√3/2)cos(x) = 0.
Step 7: Factor out sin(x) and cos(x): sin(x)(2cos(x) + 0.5) - (√3/2)cos(x) = 0.
Step 8: Set each factor equal to zero: sin(x) = 0 or (2cos(x) + 0.5) - (√3/2)cos(x) = 0.
Step 9: Solve for sin(x) = 0. There are two values of x that satisfy sin(x) = 0 within the given range x€[-180;180], these are x = -180 and x = 0.
Step 10: Now solve for (2cos(x) + 0.5) - (√3/2)cos(x) = 0. Combine like terms: 2cos(x) - (√3/2)cos(x) + 0.5 = 0.
Step 11: Simplify further: (2 - √3/2)cos(x) + 0.5 = 0.
Step 12: Subtract 0.5 from both sides: (2 - √3/2)cos(x) = -0.5.
Step 13: Divide both sides by (2 - √3/2): cos(x) = -0.5 / (2 - √3/2).
Step 14: Calculate the ratio: -0.5 / (2 - √3/2) ≈ 0.366.
Step 15: Use the inverse cosine function to find the values of x within the given range: x = arccos(0.366) ≈ 68° or x = 292°.
Hence, the solutions to the equation sin2x = cos(x+30) within the given range x€[-180;180] are x = -180, x = 0, x ≈ 68°, and x ≈ 292°.