find the angle (theta) using cofunction identites
sin(3theta-17degrees)=cos(theta+43degrees)
To find the angle (θ) using co-function identities, we need to apply the following identities:
sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)
Now, let's solve the given equation step by step:
sin(3θ - 17°) = cos(θ + 43°)
First, let's apply the co-function identity for sin(3θ - 17°):
cos(90° - (3θ - 17°)) = cos(θ + 43°)
Simplifying further:
cos(90° - 3θ + 17°) = cos(θ + 43°)
We can rewrite the equation in terms of angles only:
cos(107° - 3θ) = cos(θ + 43°)
Now, we know that the cos function is periodic with a period of 360° or 2π radians. This means that two angles whose cosines are equal must differ by an integer multiple of 360° or 2π radians. So, we can set up an equation to find the possible values of θ:
107° - 3θ = θ + 43° + 360°k
Where k is an integer representing the number of complete cycles of 360° or 2π radians.
Now, rearranging the equation:
4θ = 64° + 360°k
Divide both sides by 4:
θ = (64° + 360°k) / 4
Simplifying further:
θ = 16° + 90°k
Thus, the general solution for θ is θ = 16° + 90°k, where k is an integer. This equation represents an infinite number of solutions, as k can take any integer value.
Therefore, to find specific values of θ, you'll need to substitute different integer values for k and calculate θ. Each value of k will provide a different angle (θ) that satisfies the given equation.