Find all the solutions of the equation in the interval {o,2pi)
sec^2x+tanx=1
One of the basic identities you should have at your disposal is
sec^2 x = 1 + tan^2 x
so sec^2 x + tanx = 1
1 + tan^2 x + tanx = 1
tanx(tanx + 1) = 0
tanx = 0 or tanx = -1
if tanx = 0 then x = 0 or x = 180°
if tanx = -1, then x = 135° or x = 315°
x = 0 , π , 3π/4 , 7π/4
To solve the equation sec^2(x) + tan(x) = 1 in the interval [0, 2π], follow these steps:
Step 1: Identify the trigonometric identities associated with sec^2(x) and tan(x).
- Recall that sec^2(x) = 1/cos^2(x) and tan(x) = sin(x)/cos(x).
Step 2: Rewrite the equation using the identified trigonometric identities.
- The equation becomes 1/cos^2(x) + sin(x)/cos(x) = 1.
Step 3: Combine the terms on the left side of the equation into a single fraction.
- Multiply sin(x) by cos(x) to obtain sin(x)*cos(x)/cos(x), which simplifies to sin(x).
Therefore, the equation becomes (1 + sin(x))/cos^2(x) = 1.
Step 4: Multiply both sides of the equation by cos^2(x) to eliminate the denominator.
- The equation now becomes 1 + sin(x) = cos^2(x).
Step 5: Rearrange the equation to isolate sin(x) on one side.
- Subtract cos^2(x) from both sides of the equation to obtain sin(x) = -cos^2(x) + 1.
Step 6: Using the Pythagorean identity, replace cos^2(x) with 1 - sin^2(x).
- The equation becomes sin(x) = -(1 - sin^2(x)) + 1.
Step 7: Expand and simplify the equation further using algebraic manipulations.
- Distribute the negative sign to obtain sin(x) = -1 + sin^2(x) + 1.
- Simplify by combining like terms: sin(x) = sin^2(x).
Step 8: Set up two separate equations to solve for the possible values of sin(x).
- sin(x) = sin^2(x) can be rewritten as sin(x) - sin^2(x) = 0.
Step 9: Factor out sin(x) from the left side of the equation.
- sin(x)(1 - sin(x)) = 0.
Step 10: Set each factor equal to zero and solve for sin(x).
- sin(x) = 0 or 1 - sin(x) = 0.
Step 11: Solve for sin(x) in each equation.
- For sin(x) = 0, x = 0 and x = π.
- For 1 - sin(x) = 0, sin(x) = 1.
Since sin(x) = 1 at x = π/2, we include π/2 as a possible solution.
Therefore, the solutions to the equation sec^2(x) + tan(x) = 1 in the interval [0, 2π] are x = 0, π/2, and π.
To find all the solutions of the equation sec^2x + tanx = 1 in the interval [0, 2π), we can break it down into two parts:
1. Solve for sec^2x = 1 - tanx
2. Solve for tanx
Let's start with the first part:
1. Solve for sec^2x = 1 - tanx:
We know that sec^2x = 1/cos^2x, so the equation can be rewritten as:
1/cos^2x = 1 - tanx
Rearranging the equation:
1 = cos^2x - tanx*cos^2x
1 = cos^2x(1 - tanx)
Now we have two possibilities:
a. cos^2x = 0
This implies that cosx = 0, which gives us the solutions x = π/2 and x = 3π/2.
b. 1 - tanx = 0
Solving for tanx:
tanx = 1
The positive solutions for tanx = 1 are x = π/4 and x = 5π/4.
Now, let's move on to the second part:
2. Solve for tanx:
tanx = 1
The positive solutions for tanx = 1 are x = π/4 and x = 5π/4.
Therefore, the solutions to the equation sec^2x + tanx = 1 in the interval [0, 2π) are:
x = π/4, 5π/4, π/2, and 3π/2.