solve the equation
1+4sin(x)=4cos^2(x)
without a graphing utility
1+4sin(x)=4cos^2(x)
1+4sinx = 4(1 - sin^2 x)
4sin^2 x + 4sinx -3 = 0
(2sinx - 1)(2sinx + 3) = 0
sinx = 1/2 or sinx = -3/2
the last part is not possible since sinx is between -1 and +1
so sinx = 1/2
x must be in quadrants I or II
I know sin 30° = 1/2
so x = 30° or 150°
in radians
x = π/6 or x = 5π/6
To solve the equation 1 + 4sin(x) = 4cos^2(x), we can use trigonometric identities to simplify and solve for x. Here's the step-by-step process:
Step 1: Use the Pythagorean identity to rewrite cos^2(x) in terms of sin(x): cos^2(x) = 1 - sin^2(x).
Substituting this into our equation, we have:
1 + 4sin(x) = 4(1 - sin^2(x))
Step 2: Distribute 4 to both terms inside the parentheses:
1 + 4sin(x) = 4 - 4sin^2(x)
Rearrange the equation:
4sin^2(x) + 4sin(x) - 3 = 0
Step 3: Let's introduce a substitution: Let u = sin(x).
Now our equation becomes:
4u^2 + 4u - 3 = 0
Step 4: Solve this quadratic equation using factoring, completing the square, or applying the quadratic formula. In this case, we can use factoring.
The equation factors as:
(2u - 1)(2u + 3) = 0
Step 5: Set each factor equal to zero:
2u - 1 = 0 or 2u + 3 = 0
Solve for u:
2u = 1 or 2u = -3
u = 1/2 or u = -3/2
Step 6: Remember that u = sin(x), so substitute back:
sin(x) = 1/2 or sin(x) = -3/2
Step 7: Solve for x by taking the inverse sine (sin^-1) of both sides:
x = sin^-1(1/2) or x = sin^-1(-3/2)
Step 8: Use your calculator in radian mode to find the solutions:
x ≈ π/6 + 2πn or x ≈ 11π/6 + 2πn, where n is an integer.
So, the solutions to the equation 1 + 4sin(x) = 4cos^2(x) are approximately:
x ≈ π/6 + 2πn or x ≈ 11π/6 + 2πn, where n is an integer.