Question

question 3 (1 point)
determine the value of θ to the nearest degree if cotθ = 0.75.
a) 45°
b) 37°
c) 53°
d) 42°

Explanation:

Step1: Recall the relationship between cotangent and arccotangent

We know that if \(\cot\theta = x\), then \(\theta=\text{arccot}(x)\). Also, \(\cot\theta=\frac{\cos\theta}{\sin\theta}\) and \(\text{arccot}(x)=\tan^{- 1}(\frac{1}{x})\) for \(x>0\). Given \(\cot\theta = 0.75\), then \(\theta=\text{arccot}(0.75)\). Using the identity \(\text{arccot}(x)=\tan^{-1}(\frac{1}{x})\), we have \(\theta=\tan^{-1}(\frac{1}{0.75})\).

Step2: Calculate \(\tan^{-1}(\frac{4}{3})\)

We know that \(\frac{1}{0.75}=\frac{4}{3}\). Using a calculator, \(\tan^{-1}(\frac{4}{3})\approx53.13^{\circ}\). Rounding \(53.13^{\circ}\) to the nearest degree gives \(53^{\circ}\).

Answer:

C. \(53^{\circ}\)