Final answer:
To find the values of theta between 0 and 27 for which cot theta = sqrt(-13), we can use the unit circle and the definition of cotangent. Substitute the value of tan(theta) into the equation, and rearrange to find cos(theta) = -sqrt(13) sin(theta). By using the unit circle, we find that theta = 14.4784 and theta = 23.5216 satisfy the equation.
Step-by-step explanation:
To find the values of θ between 0 and 27 for which cot θ = √-13, we can use the unit circle and the definition of cotangent.
The unit circle is a circle with a radius of 1 unit centered at the origin (0,0) in a coordinate plane. The x-coordinate represents the cosine value, and the y-coordinate represents the sine value.
The cotangent θ is defined as the reciprocal of the tangent θ, which is equal to 1/tan θ. So, we need to find the values of θ where 1/tan θ = √-13.
Since tan θ = sin θ/cos θ, we can substitute this value into the equation. We end up with 1/(sin θ/cos θ) = √-13. Rearranging this equation, we get cos θ = -√13 sin θ.
Now we can use the unit circle to find the angles θ that satisfy this equation. Let's look for values between 0 and 27. We find that the angles θ = 14.4784 and θ = 23.5216 satisfy the equation cos θ = -√13 sin θ.
Therefore, the values of θ between 0 and 27 for which cot θ = √-13 are 14.4784 and 23.5216.
Learn more about cotangent