2sec^2x - 3tanx - 5 = 0
I got
x= 1.24 radians
x= 4.38 radians
x= -0.74 radians
x= -3.88 radians
trying one of them
my calculator is set for degrees
using 4.38 rad = 251 degrees
sec 251 = -3.07
2 sec^2 251 = 18.87
tan 251 = 2.9
-3 tan 251 = -8.71
18.87 -8.71 - 5 = 5.15
hmmm :(
2sec^2x - 3tanx - 5 = 0
2 + 2tan^2x - 3tanx - 5 = 0
2tan^2x - 3tanx - 3 = 0
tanx = (3±√33)/4 = -0.686, 2.1861
x = -0.601+nπ, 1.142+nπ
So, I get
x = 1.142, 2.541, 4.283, 5.682
Too bad you didn't show us any of your work.
To solve the equation 2sec^2x - 3tanx - 5 = 0, you need to find the values of x that satisfy the equation. Here's how you can solve it step by step:
Step 1: Recall the trigonometric identities:
- sec^2x = 1 + tan^2x
- tanx = sinx/cosx
Step 2: Substitute the identities into the equation:
2(1 + tan^2x) - 3(sinx/cosx) - 5 = 0
Step 3: Simplify the equation:
2 + 2tan^2x - 3sinx/cosx - 5 = 0
2tan^2x - 3sinx/cosx - 3 = 0
Step 4: Multiply the entire equation by cosx to eliminate the denominator:
2tan^2x*cosx - 3sinx - 3cosx = 0
Step 5: Rewrite tan^2x as sin^2x/cos^2x:
2(sin^2x/cos^2x)*cosx - 3sinx - 3cosx = 0
Step 6: Simplify the equation further:
2sin^2x/cosx - 3sinx - 3cosx = 0
Step 7: Multiply the entire equation by cosx to eliminate the denominator:
2sin^2x - 3sinxcosx - 3cos^2x = 0
Step 8: Recognize that sin^2x = 1 - cos^2x:
2(1 - cos^2x) - 3sinxcosx - 3cos^2x = 0
Step 9: Simplify the equation:
2 - 2cos^2x - 3sinxcosx - 3cos^2x = 0
-5cos^2x - 3sinxcosx + 2 = 0
Step 10: Rearrange the equation to make it easier to factor:
5cos^2x + 3sinxcosx - 2 = 0
Step 11: Factor the equation:
(5cosx - 2)(cosx + 1) = 0
Step 12: Set each factor equal to zero and solve separately:
5cosx - 2 = 0
cosx + 1 = 0
Step 13: Solve each equation:
cosx = 2/5
cosx = -1
Step 14: Find the corresponding values of x:
For cosx = 2/5,
x = arccos(2/5)
x ≈ 1.24 radians
For cosx = -1,
x = arccos(-1)
x ≈ 3.14 radians (π radians)
So, the solutions to the equation are x ≈ 1.24 radians, x ≈ 3.14 radians (π radians).