Question

A) Sin(x+π/5)=√2/2 б) cos(3x-π/6)=-1

Answer 1

А)

$sin(x+ \frac{ \pi }{5} )= \frac{ \sqrt{2} }{2}$

$x+ \frac{ \pi }{5}= (-1)^narcsin\frac{ \sqrt{2} }{2} + \pi n,$  $n$ ∈  $Z$

$x+ \frac{ \pi }{5}= (-1)^n{ \frac{ \pi }{4} + \pi n,$  $n$ ∈  $Z$

$x= (-1)^n{ \frac{ \pi }{4} - \frac{ \pi }{5} + \pi n,$  $n$ ∈  $Z$

б)

$cos(3x- \frac{ \pi }{6} )=-1$

$3x- \frac{ \pi }{6}= \pi +2 \pi n,$  $n$ ∈  $Z$

$3x= \pi +\frac{ \pi }{6}+2 \pi n,$  $n$ ∈  $Z$

$3x=\frac{ 7\pi }{6}+2 \pi n,$  $n$ ∈  $Z$

$x=\frac{ 7\pi }{18}+ \frac{2 \pi n}{3} ,$  $n$ ∈  $Z$

Answer 2

To solve the given trigonometric equations, we'll find the values of 'x' that satisfy each equation. Let's solve them one by one:

a) Sin(x + π/5) = √2/2

First, we'll find the general solution for the equation by isolating 'x' on one side of the equation:

sin(x + π/5) = √2/2

To find the general solution for sin(theta) = √2/2, we look at the unit circle. The sine function is positive in the first and second quadrants when the angle is π/4 or 3π/4:

sin(π/4) = √2/2 sin(3π/4) = √2/2

Now, to find the general solution for 'x':

x + π/5 = π/4 + 2πn or x + π/5 = 3π/4 + 2πn

where 'n' is an integer representing the number of full circles we go around the unit circle.

Now, let's solve for 'x':

For the first case (π/4 + 2πn):

x = π/4 - π/5 + 2πn x = (5π - 4π)/(20) + 2πn x = π/20 + 2πn

For the second case (3π/4 + 2πn):

x = 3π/4 - π/5 + 2πn x = (15π - 4π)/(20) + 2πn x = 11π/20 + 2πn

So, the general solutions for the equation sin(x + π/5) = √2/2 are:

x = π/20 + 2πn or x = 11π/20 + 2πn

where 'n' is an integer.

b) cos(3x - π/6) = -1

Again, let's find the general solution for the equation:

cos(3x - π/6) = -1

To find the general solution for cos(theta) = -1, we look at the unit circle. The cosine function is equal to -1 when the angle is π:

cos(π) = -1

Now, to find the general solution for 'x':

3x - π/6 = π + 2πn

where 'n' is an integer representing the number of full circles we go around the unit circle.

Now, let's solve for 'x':

3x = π + π/6 + 2πn x = (π + π/6 + 2πn) / 3 x = (6π + π + 12πn) / 18 x = (7π + 12πn) / 18 x = (7π/18) + (2πn/3)

So, the general solution for the equation cos(3x - π/6) = -1 is:

x = (7π/18) + (2πn/3)

where 'n' is an integer.

Remember, these solutions represent all possible values of 'x' that satisfy the given trigonometric equations. If you need a specific range for 'x' or numerical values, you can substitute different values of 'n' into the general solutions to get specific solutions for 'x'.