Solve for x in the interval 0<=x<360:
1. 2sin2x+cosx =0
2. cos2x=-2sinx
3. tanx=2sinx
4. 3cos2x+cosx+2=0
1. To solve the equation 2sin2x + cosx = 0, we can use the following steps:
Step 1: Rewrite sin2x using the double-angle identity sin2x = 2sinxcosx.
--> 2(2sinxcosx) + cosx = 0
Step 2: Simplify the equation:
--> 4sinxcosx + cosx = 0
Step 3: Factor out cosx from the equation:
--> cosx(4sinx + 1) = 0
Step 4: Set each factor equal to zero and solve for x:
--> cosx = 0
--> 4sinx + 1 = 0
For the first factor cosx = 0, x can be any angle where cosx is zero. The values of x that satisfy cosx = 0 in the interval 0 <= x < 360 are: 90° and 270°.
For the second factor 4sinx + 1 = 0, rearrange the equation:
--> 4sinx = -1
--> sinx = -1/4
Using a calculator or reference table, find the values of x that satisfy sinx = -1/4 in the interval 0 <= x < 360. Solving for x, you will find x = 210° and x = 330°.
So, in the interval 0 <= x < 360, the solutions for the equation 2sin2x + cosx = 0 are x = 90°, x = 210°, x = 270°, and x = 330°.
2. To solve the equation cos2x = -2sinx, we can use the following steps:
Step 1: Rewrite cos2x using the double-angle identity cos2x = cos^2(x) - sin^2(x).
--> cos^2(x) - sin^2(x) = -2sinx
Step 2: Rewrite sin^2(x) as 1 - cos^2(x) using the Pythagorean identity for sine and cosine.
--> cos^2(x) - (1 - cos^2(x)) = -2sinx
--> 2cos^2(x) - 1 = -2sinx
Step 3: Rearrange the equation to get cos^2(x) alone:
--> 2cos^2(x) = -2sinx + 1
Step 4: Remember that cos^2(x) = 1 - sin^2(x) using the Pythagorean identity, substitute this in the equation:
--> 2(1 - sin^2(x)) = -2sinx + 1
--> 2 - 2sin^2(x) = -2sinx + 1
Step 5: Move all terms to one side of the equation:
--> 2sin^2(x) - 2sinx - 1 = 0
Step 6: This quadratic equation can now be solved using various methods. One way is to factor:
--> (2sinx + 1)(sinx - 1) = 0
Step 7: Set each factor equal to zero and solve for x:
--> 2sinx + 1 = 0
--> sinx - 1 = 0
For the first factor 2sinx + 1 = 0, solve for sinx:
--> 2sinx = -1
--> sinx = -1/2
Using a calculator or reference table, find the values of x that satisfy sinx = -1/2 in the interval 0 <= x < 360. Solving for x, you will find x = 210° and x = 330°.
For the second factor sinx - 1 = 0, solve for sinx:
--> sinx = 1
Using a calculator or reference table, find the value of x that satisfies sinx = 1 in the interval 0 <= x < 360. Solving for x, you will find x = 90°.
So, in the interval 0 <= x < 360, the solutions for the equation cos2x = -2sinx are x = 90°, x = 210°, and x = 330°.
3. To solve the equation tanx = 2sinx, we can use the following steps:
Step 1: Rewrite tanx using the definition of tangent as sinx/cosx:
--> sinx/cosx = 2sinx
Step 2: Multiply both sides of the equation by cosx to eliminate the denominator:
--> sinx = 2sinxcosx
Step 3: Simplify the equation:
--> 1 = 2cosx
Step 4: Divide by 2:
--> 1/2 = cosx
Step 5: Using a calculator or reference table, find the values of x that satisfy cosx = 1/2 in the interval 0 <= x < 360. Solving for x, you will find x = 60° and x = 300°.
So, in the interval 0 <= x < 360, the solutions for the equation tanx = 2sinx are x = 60° and x = 300°.
4. To solve the equation 3cos2x + cosx + 2 = 0, we can use the following steps:
Step 1: Rewrite cos2x using the double-angle identity cos2x = cos^2(x) - sin^2(x).
--> 3(cos^2(x) - sin^2(x)) + cosx + 2 = 0
Step 2: Rearrange the equation to get cos^2(x) alone:
--> 3cos^2(x) - 3sin^2(x) + cosx + 2 = 0
Step 3: Remember that cos^2(x) = 1 - sin^2(x) using the Pythagorean identity, substitute this in the equation:
--> 3(1 - sin^2(x)) - 3sin^2(x) + cosx + 2 = 0
--> 3 - 3sin^2(x) - 3sin^2(x) + cosx + 2 = 0
Step 4: Combine like terms:
--> -6sin^2(x) + cosx + 5 = 0
Step 5: Set each term equal to zero to solve for x:
--> -6sin^2(x) = -cosx - 5
Step 6: Square both sides of the equation to eliminate the square term:
--> 36sin^4(x) = cos^2(x) + 10cosx + 25
Step 7: Use the identity sin^2(x) = 1 - cos^2(x), substitute this in the equation:
--> 36(1 - cos^2(x))^2 = cos^2(x) + 10cosx + 25
Step 8: Expand and simplify the equation:
--> 36(1 - 2cos^2(x) + cos^4(x)) = cos^2(x) + 10cosx + 25
Step 9: Rearrange the equation to get a quadratic equation in terms of cosx:
--> 36cos^4(x) - 72cos^2(x) + (36 + cos^2(x) + 10cosx + 25) = 0
--> 36cos^4(x) - 72cos^2(x) + cos^2(x) + 10cosx + 61 = 0
Step 10: Simplify the equation:
--> 36cos^4(x) - 71cos^2(x) + 10cosx + 61 = 0
Solving this quartic equation requires more advanced techniques beyond the scope of this explanation. You can use numerical methods or software to find the approximate solutions.
So, in the interval 0 <= x < 360, the solutions for the equation 3cos2x + cosx + 2 = 0 can be found using numerical methods or software.