Solve 3cosx+1=2secx
To solve the equation 3cos(x) + 1 = 2sec(x), we need to first simplify the equation using trigonometric identities and then solve for x.
Step 1: Rewrite sec(x) in terms of cos(x).
Recall that sec(x) is the reciprocal of cos(x), therefore, sec(x) = 1/cos(x).
Step 2: Substitute the value of sec(x) in the equation.
The equation becomes: 3cos(x) + 1 = 2/cos(x).
Step 3: Multiply both sides of the equation by cos(x) to eliminate the fraction.
Doing this will give us: 3cos^2(x) + cos(x) = 2.
Step 4: Simplify the equation.
Rearrange the equation: 3cos^2(x) + cos(x) - 2 = 0.
Step 5: Factor the quadratic equation (if possible).
Let's try factoring first: (3cos(x) - 1)(cos(x) + 2) = 0.
Step 6: Solve for x.
To find the values of x that satisfy the equation, we set each factor equal to zero and solve for x separately.
Setting 3cos(x) - 1 = 0, we get:
3cos(x) = 1
cos(x) = 1/3
x = arccos(1/3) ≈ 1.23096
Setting cos(x) + 2 = 0, we get:
cos(x) = -2
Since the range of cos(x) is [-1, 1], there are no real solutions for this case.
Therefore, the solution for the equation 3cos(x) + 1 = 2sec(x) is x ≈ 1.23096 (rounded to five decimal places).
To solve the equation 3cos(x) + 1 = 2sec(x), we will first start by rewriting the equation in terms of sine and cosine:
3cos(x) + 1 = 2sec(x)
Since sec(x) is the reciprocal of cos(x), we can rewrite it as:
3cos(x) + 1 = 2/cos(x)
Now, let's multiply both sides of the equation by cos(x) to eliminate the denominator:
3cos(x) * cos(x) + 1 * cos(x) = 2
Simplifying further:
3cos^2(x) + cos(x) = 2
Next, let's use the identity cos^2(x) = 1 - sin^2(x) to rewrite the equation:
3(1 - sin^2(x)) + cos(x) = 2
Expanding and rearranging terms:
3 - 3sin^2(x) + cos(x) = 2
Rearranging further:
3 - 2 - 3sin^2(x) + cos(x) = 0
Combining like terms:
-3sin^2(x) + cos(x) + 1 = 0
Now, let's solve for sin(x):
-3sin^2(x) + cos(x) + 1 = 0
We know that sin^2(x) = 1 - cos^2(x). Substituting this into the equation:
-3(1 - cos^2(x)) + cos(x) + 1 = 0
Expanding and rearranging:
-3 + 3cos^2(x) + cos(x) + 1 = 0
Combining like terms:
3cos^2(x) + cos(x) - 2 = 0
Now, let's solve this quadratic equation for cos(x) by factoring or using the quadratic formula. Once we find the values of cos(x), we can substitute them back into the original equation to solve for x.