1. express tan 11pi/6 in terms of a positive acute angle
2. solve for x : sin(4x-7) = cos 17
thank you!
tan pi/3
x= (approx)1.68
pi/6
1. To express tan(11π/6) in terms of a positive acute angle, we need to find an equivalent angle within the range of 0 to π/2 (or 0 to 90 degrees).
We know that the tangent function has a period of π, which means that tan(x) = tan(x + π) for any value of x.
In this case, 11π/6 is already greater than π, so we can subtract π from it to find an equivalent angle within the desired range:
11π/6 - π = 5π/6
Now we have an angle of 5π/6, which is still outside the range of 0 to π/2. To further simplify it, we can subtract π/2 to obtain an angle in the first quadrant:
5π/6 - π/2 = (5π - 3π)/6 = 2π/6 = π/3
So, tan(11π/6) is equivalent to tan(π/3) in terms of a positive acute angle.
2. To solve sin(4x - 7) = cos(17) for x, we need to isolate x on one side of the equation.
First, let's rearrange the equation to have sin(4x - 7) - cos(17) = 0:
sin(4x - 7) - cos(17) = 0
Next, we can use the identity sin(a) - cos(b) = sqrt(2) * sin(a - b - π/4) to simplify the equation:
sin(4x - 7) - cos(17) = sqrt(2) * sin(4x - 7 - 17 - π/4)
Now we have:
sqrt(2) * sin(4x - 24 - π/4) = 0
To solve this equation, we set the expression inside the sine function equal to zero:
4x - 24 - π/4 = 0
Simplifying, we have:
4x - 24 = π/4
Adding 24 to both sides:
4x = π/4 + 24
Now divide by 4:
x = (π/4 + 24) / 4
This gives us the solution for x in terms of π. If you need a numerical approximation, you can substitute the value of π (approximately 3.14159) into the equation to find an approximate value for x.