Question 59 refers to the following: Use exact numerical values to show the gi statement is true for the indicated value of all work.] 59) $$ an heta=\frac{\sec heta}{\operatorname{ccs} heta} ; heta=\frac{\pi}{3} $$

Question

Question 59 refers to the following: Use exact numerical values to show the gi statement is true for the indicated value of all work.] 59) $$ 	an 	heta=\frac{\sec 	heta}{\operatorname{ccs} 	heta} ; 	heta=\frac{\pi}{3} $$

UpStudy AI · Accepted Answer

To verify the statement

$$
\tan \theta = \frac{\sec \theta}{\csc \theta} \quad \text{for} \quad \theta = \frac{\pi}{3},
$$

we'll use exact trigonometric values.

### Step 1: Recall Exact Values at $$\theta = \frac{\pi}{3}$$
$$
\theta = \frac{\pi}{3} \quad \text{(60 degrees)}
$$
$$
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
$$
$$
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
$$
$$
\tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}
$$
$$
\sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2
$$
$$
\csc\left(\frac{\pi}{3}\right) = \frac{1}{\sin\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
$$

### Step 2: Compute $$\frac{\sec \theta}{\csc \theta}$$
$$
\frac{\sec\left(\frac{\pi}{3}\right)}{\csc\left(\frac{\pi}{3}\right)} = \frac{2}{\frac{2\sqrt{3}}{3}} = 2 \times \frac{3}{2\sqrt{3}} = \frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}
$$

### Step 3: Compare Both Sides
$$
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
$$
$$
\frac{\sec\left(\frac{\pi}{3}\right)}{\csc\left(\frac{\pi}{3}\right)} = \sqrt{3}
$$

Since both sides equal $$\sqrt{3}$$, the statement is true for $$\theta = \frac{\pi}{3}$$.

**Conclusion:**
$$
\tan\left(\frac{\pi}{3}\right) = \frac{\sec\left(\frac{\pi}{3}\right)}{\csc\left(\frac{\pi}{3}\right)} = \sqrt{3}
$$

This confirms that the given identity holds for $$\theta = \frac{\pi}{3}$$.
$$ \tan\left(\frac{\pi}{3}\right) = \frac{\sec\left(\frac{\pi}{3}\right)}{\csc\left(\frac{\pi}{3}\right)} = \sqrt{3} $$ The statement is true for $$\theta = \frac{\pi}{3}$$.

Question 59 refers to the following: Use exact numerical values to show the gi statement is true for the indicated value of all work.] 59) \( \tan \theta=\frac{\sec \theta}{\operatorname{ccs} \theta} ; \theta=\frac{\pi}{3} \)

Solución de tutoría real

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Trigonometry Questions