Solve the equations
secΘ - 1 = (√2 - 1)tanΘ
To solve the equation secΘ - 1 = (√2 - 1)tanΘ, we can start by simplifying the equation using trigonometric identities.
Recall the following identities:
- secΘ = 1/cosΘ
- tanΘ = sinΘ/cosΘ
Now, let's rewrite the equation using these identities:
1/cosΘ - 1 = (√2 - 1)(sinΘ/cosΘ)
Next, let's clear the fraction by multiplying both sides of the equation by cosΘ:
1 - cosΘ = (√2 - 1)sinΘ
Now, we can express sinΘ in terms of cosΘ using the Pythagorean identity:
sin^2Θ + cos^2Θ = 1
sin^2Θ = 1 - cos^2Θ
sinΘ = √(1 - cos^2Θ)
Substituting this into our equation, we have:
1 - cosΘ = (√2 - 1)√(1 - cos^2Θ)
Let's square both sides of the equation to eliminate the square root:
(1 - cosΘ)^2 = (√2 - 1)^2(1 - cos^2Θ)
Expanding both sides:
1 - 2cosΘ + cos^2Θ = (3 - 2√2) - (2 - 2√2)cos^2Θ
Rearranging the terms:
cos^2Θ - (2 - 2√2)cos^2Θ = (3 - 2√2) - 1 + 2cosΘ
Combining like terms:
(-1 + 2√2 - 1)cos^2Θ + 2cosΘ + 2 - 3 + 2√2 = 0
Simplifying:
(-2 + 2√2)cos^2Θ + 2cosΘ - 1 + 2√2 = 0
Now, we have a quadratic equation in terms of cosΘ. We can solve this quadratic equation by factoring, using the quadratic formula, or by graphing.
I hope this explanation helps you solve the equation!
(1/cos x)-1 = (sqrt 2 - 1)(sin x/cos x)
1 - cos x = sin x sqrt 2 - sin x
1 - cos x + sin x = sin x sqrt 2
I want sin x = cos x so try x = 45 degrees
1 = (1/sqrt 2)sqrt 2 sure enough
so x = 45 degrees or pi/4 radians