Solve.
(A) 2sin^2 x = 1
(B) tan^2 x = 3
did you try first?
a) sin^2x = 1/2
sinx = sqrt(2)/2
b) tanx = sqrt(3)
hope you can do it from here.
Sure! Let's solve each equation step by step.
(A) 2sin^2(x) = 1
To solve this equation, we need to isolate sin^2(x). First, divide both sides of the equation by 2:
sin^2(x) = 1/2
Now, take the square root of both sides to solve for sin(x):
sin(x) = ±√(1/2)
To find the exact values of sin(x), note that sin is positive in the first and second quadrants. Therefore:
sin(x) = √(1/2) or sin(x) = -√(1/2)
Simplifying the square root:
sin(x) = √(1/2) or sin(x) = -√(1/2)
sin(x) = √2/2 or sin(x) = -√2/2
The solutions for x are the angles whose sine value is √2/2 or -√2/2. These angles can be found from the unit circle or by using trigonometric values. The solutions are:
x = π/4 + 2πn or x = 3π/4 + 2πn, where n is an integer.
(B) tan^2 x = 3
To solve this equation, let's isolate tan(x). Take the square root of both sides:
tan(x) = ±√3
Since tan is positive in the first and third quadrants:
tan(x) = √3 or tan(x) = -√3
To find the exact values of tan(x), you can refer to trigonometric values or the unit circle. The solutions for x are the angles whose tangent value is √3 or -√3. The solutions are:
x = π/3 + πn or x = 2π/3 + πn, where n is an integer.