Question

1. Sin x=12. cos x=-1/23. 2 sin x-1=04. tgx=1/корень 35. cos3x=1/2

Answer 1

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Answer 2

To solve the given trigonometric equations, we'll find the values of x that satisfy each equation.

1. sin(x) = 12 There is no real number whose sine value is greater than 1 or less than -1, so this equation has no solution.

2. cos(x) = -1/23 To find x, we take the arccosine of -1/23: x = arccos(-1/23)

Using a calculator, we get the approximate value of x as -1.5668 radians (or about -89.8244 degrees).

3. 2sin(x) - 1 = 0 Add 1 to both sides and then divide by 2: 2sin(x) = 1 sin(x) = 1/2

To find x, we take the arcsine of 1/2: x = arcsin(1/2)

Using a calculator, we get the two possible values of x as π/6 radians (or about 30 degrees) and 5π/6 radians (or about 150 degrees).

4. tan(x) = 1/√35 To find x, we take the arctangent of 1/√35: x = arctan(1/√35)

Using a calculator, we get the approximate value of x as 0.1860 radians (or about 10.67 degrees).

5. cos(3x) = 1/2 To find x, we first need to find the solutions to cos(x) = 1/2, and then divide them by 3.

cos(x) = 1/2 The solutions for x in the range [0, 2π) are x = π/3 and x = 5π/3.

Now, divide these solutions by 3 to find the solutions for 3x: For x = π/3, 3x = π (approximately 3.1416 radians). For x = 5π/3, 3x = 5π (approximately 15.7079 radians).

So, the solutions for the equation cos(3x) = 1/2 are 3.1416 radians and 15.7079 radians (approximately 180 and 900 degrees, respectively).

Please note that some of these equations have no real solutions, and others have multiple solutions due to the periodic nature of trigonometric functions. Be sure to check the domains and ranges of the functions involved when solving such equations.