find the exact value of the trigonometric function at the given real number.\n(a) \\( \\sin \\left( - \\frac…

Explanation:

Step1: Use the property of sine function

The sine function is odd, i.e., (\sin(-x)=-\sin(x)). So, (\sin\left(-\frac{\pi}{2}\right)=-\sin\left(\frac{\pi}{2}\right)). Since (\sin\left(\frac{\pi}{2}\right) = 1), then (\sin\left(-\frac{\pi}{2}\right)=- 1).

Step2: Use the property of cosine function

The cosine function is even, i.e., (\cos(-x)=\cos(x)). So, (\cos\left(-\frac{\pi}{2}\right)=\cos\left(\frac{\pi}{2}\right)). Since (\cos\left(\frac{\pi}{2}\right)=0), then (\cos\left(-\frac{\pi}{2}\right)=0).

Step3: Use the definition of cotangent function

The cotangent function is (\cot(x)=\frac{\cos(x)}{\sin(x)}). So, (\cot\left(-\frac{\pi}{2}\right)=\frac{\cos\left(-\frac{\pi}{2}\right)}{\sin\left(-\frac{\pi}{2}\right)}). Substituting the values from step1 and step2: (\cot\left(-\frac{\pi}{2}\right)=\frac{0}{-1}=0).

Answer:

(a) (-1) (b) (0) (c) (0)