Find all real solutions to four decimal places to the equation sec^2(x)+ 3 tan(x)=5
recall that sec^2 = 1+tan^2, so we have
tan^2(x) + 3tan(x) - 4 = 0
(tan(x)+4)(tan(x)-1) = 0
tan(x) = 1 or -4
So, there will be 4 solutions, one in each quadrant.
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To find all real solutions to the equation sec^2(x) + 3 tan(x) = 5, we can use a combination of trigonometric identities and algebraic manipulation.
Let's start by rewriting sec^2(x) in terms of sin(x) and cos(x). The identity sec^2(x) = 1 + tan^2(x) states that the square of the secant function is equal to one plus the square of the tangent function.
Substituting this identity into the original equation, we get (1 + tan^2(x)) + 3 tan(x) = 5.
Rearranging the terms and simplifying, we have tan^2(x) + 3 tan(x) - 4 = 0.
Now, let's solve this quadratic equation for tan(x).
We can factorize the quadratic equation as (tan(x) + 4)(tan(x) - 1) = 0.
Setting each factor equal to zero, we have two possibilities: tan(x) + 4 = 0 or tan(x) - 1 = 0.
For tan(x) + 4 = 0, we subtract 4 from both sides and find that tan(x) = -4.
For tan(x) - 1 = 0, we add 1 to both sides and find that tan(x) = 1.
Now, we need to find the corresponding values of x that satisfy these solutions for tan(x).
To find the values of x for tan(x) = -4, we can use the inverse tangent function (also known as arctan or tan^(-1)). We find that x = arctan(-4) ≈ -1.3258 radians or x ≈ 1.8158 radians.
To find the values of x for tan(x) = 1, we again use the inverse tangent function. We find that x = arctan(1) ≈ 0.7854 radians or x ≈ 2.3562 radians.
Hence, the real solutions to the equation sec^2(x) + 3 tan(x) = 5, rounded to four decimal places, are approximately x ≈ -1.3258, 0.7854, 1.8158, and 2.3562 radians.