Question

use the trigonometric function values of the quadrantal angles to evaluate.
7 cot 270° + 3 cos 0° + 5( csc 90° )²

7 cot 270° + 3 cos 0° + 5( csc 90° )² = □
(simplify your answer. type an integer or a fraction.)

Explanation:

Step1: Recall trigonometric values of quadrantal angles

• $\cot 270^{\circ}$: $\cot\theta=\frac{\cos\theta}{\sin\theta}$, $\cos 270^{\circ}=0$, $\sin 270^{\circ}=-1$, so $\cot 270^{\circ}=\frac{0}{-1}=0$.
• $\cos 0^{\circ}=1$.
• $\csc 90^{\circ}$: $\csc\theta=\frac{1}{\sin\theta}$, $\sin 90^{\circ}=1$, so $\csc 90^{\circ}=\frac{1}{1}=1$, then $(\csc 90^{\circ})^2 = 1^2 = 1$.

Step2: Substitute values into the expression

Substitute $\cot 270^{\circ}=0$, $\cos 0^{\circ}=1$, and $(\csc 90^{\circ})^2 = 1$ into $7\cot 270^{\circ}+3\cos 0^{\circ}+5(\csc 90^{\circ})^2$:
\[

$$\begin{align*} &7\times0 + 3\times1 + 5\times1\\ =&0 + 3 + 5 \end{align*}$$

\]

Step3: Simplify the result

\[
0 + 3 + 5 = 8
\]

Answer:

8