determine the general solution...2.sin(2x+30)=tan 135
The 135 suggests that you are working in degrees, so
2 sin(2x+30) = tan 135°
2sin(2x+30) = -1
sin(2x + 30) = -1/2
sine is negative in III and IV
so 2x+30 = 210 or 2x+30 = 330
2x = 180 or 2x = 300
x = 90 or x = 150
The period of sin(2x+30) is 360°/2 or 180°
so general solutions are:
90° + 180k° or 150° + 180k°
To determine the general solution for the equation 2sin(2x+30) = tan 135, we need to follow a step-by-step approach.
Step 1: Convert the equation to trigonometric functions with the same argument.
In this case, the tangent of 135 degrees can be expressed as a sine or cosine function using the trigonometric identity tan θ = sin θ / cos θ.
So, tan 135 = sin 135 / cos 135.
Step 2: Simplify the equation.
2sin(2x+30) = tan 135 can be rewritten as 2sin(2x+30) = sin 135 / cos 135.
Step 3: Multiply both sides of the equation by cos 135 to eliminate the denominator.
2sin(2x+30) * cos 135 = sin 135.
Step 4: Use the trigonometric identity sin(α + β) = sin α * cos β + cos α * sin β.
In this case, α = 2x and β = 30.
2sin(2x)cos(135) + 2cos(2x)sin(135) = sin 135.
Step 5: Substitute the values of sin 135 and cos 135.
2sin(2x)cos(135) + 2cos(2x)sin(135) = sqrt(2)/2.
Step 6: Simplify and combine like terms.
sqrt(2)sin(2x) - sqrt(2)cos(2x) = sqrt(2)/2.
Step 7: Divide both sides of the equation by sqrt(2) to isolate the trigonometric functions.
sin(2x) - cos(2x) = 1/2.
Step 8: Apply the trigonometric identity sin(α - β) = sin α * cos β - cos α * sin β.
In this case, α = 2x and β = π/4.
sin(2x - π/4) = 1/2.
Step 9: Find the solutions for the simplified equation.
To determine the general solution, we solve for 2x - π/4 first.
2x - π/4 = π/6 + 2nπ or 2x - π/4 = 5π/6 + 2nπ, where n is an integer.
Step 10: Solve for x.
For the first case, 2x - π/4 = π/6 + 2nπ.
Rearrange the equation to isolate x: 2x = π/6 + π/4 + 2nπ.
Divide both sides by 2: x = (π/6 + π/4 + 2nπ)/2.
Simplifying further, x = π/12 + π/8 + nπ.
For the second case, 2x - π/4 = 5π/6 + 2nπ.
Following similar steps as before, we get x = (5π/6 + π/4 + 2nπ)/2.
Simplifying further, x = 11π/24 + nπ/2.
So, the general solution for the equation 2sin(2x+30) = tan 135 is:
x = π/12 + π/8 + nπ or x = 11π/24 + nπ/2, where n is an integer.