Answer: Using the cofunction identity for tangent and cotangent:
cot(θ) = 1/tan(θ)
We can rewrite the given equation as:
cot(5θ - 32°) = 1/tan(θ + 26°)
Next, using the identity for the tangent of the sum of two angles:
tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))
We can rewrite the right side of the equation as:
1/tan(θ + 26°) = tan(90° - (θ + 26°)) = tan(64° - θ)
Substituting this back into the original equation:
cot(5θ - 32°) = tan(64° - θ)
Using the identity for the cotangent and tangent of the difference of two angles:
cot(a - b) = (cot(a)cot(b) - 1)/(cot(b) - cot(a))
tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)tan(b))
We can rewrite the equation as:
(cot(5θ)cot(32°) - 1)/(cot(32°) - cot(5θ)) = (tan(64°)tan(θ) - tan(θ))/(1 + tan(64°)tan(θ))
Simplifying both sides:
(cot(5θ)cot(32°) - 1)/(cot(32°) - cot(5θ)) = (sin(64°)sin(θ))/(cos(64°)cos(θ) + sin(64°)sin(θ))
Cross-multiplying and simplifying:
cos(64°)cos(θ)cot(5θ) - sin(64°)sin(θ)cot(5θ) = -sin(64°)sin(θ)
cos(64°)cos(θ)cot(5θ) = sin(64°)sin(θ)(cot(5θ) + 1)
cos(64°)cos(θ)cot(5θ) = sin(64°)sin(θ)csc(5θ)
cos(64°)cos(θ) = sin(64°)sin(θ)sin(5θ)/cos(5θ)
cos(64°)cos(θ)cos(5θ) = sin(64°)sin(θ)sin(5θ)
Using the identity for the cosine of the sum of two angles:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
We can rewrite the equation as:
cos(64° + θ - 5θ) = 0
cos(64° - 4θ) = 0
64° - 4θ = 90° + k(180°) or 64° - 4θ = 270° + k(180°) where k is an integer
Solving for θ:
64° - 4θ = 90° + k(180°)
-4θ = 26° + k(180°)
θ = -(26°/4) - (k/4)(180°)
θ = -6.5° - 45°k
or
64° - 4θ = 270° + k(180°)
-4θ = 206° + k(180°)
θ = -(206°/4) - (k/4)(180°)
θ = -51.5° - 45°k
Therefore, there are two sets of solutions for θ, given by:
θ = -6.5
Explanation: