Find the exact value of the expression without using a calculator.

Ask by Peterson Lee. in the United States

Jan 20,2025

Question

Find the exact value of the expression without using a calculator. $$ \sec \left(\frac{2 \pi}{3}\right)+\sin \left(\frac{\pi}{4}\right)=\square $$

UpStudy AI · Accepted Answer

To find the exact value of the expression $$ \sec \left(\frac{2 \pi}{3}\right) + \sin \left(\frac{\pi}{4}\right) $$, let's evaluate each trigonometric function step by step.

1. **Calculate $$ \sec \left(\frac{2 \pi}{3}\right) $$:**

The secant function is the reciprocal of the cosine function:
   $$
   \sec\left(\frac{2\pi}{3}\right) = \frac{1}{\cos\left(\frac{2\pi}{3}\right)}
   $$
   
   Now, find $$ \cos\left(\frac{2\pi}{3}\right) $$:
   $$
   \cos\left(\frac{2\pi}{3}\right) = \cos\left(\pi - \frac{\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}
   $$
   
   Therefore:
   $$
   \sec\left(\frac{2\pi}{3}\right) = \frac{1}{-\frac{1}{2}} = -2
   $$

2. **Calculate $$ \sin \left(\frac{\pi}{4}\right) $$:**
   $$
   \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
   $$

3. **Combine the results:**
   $$
   \sec\left(\frac{2\pi}{3}\right) + \sin\left(\frac{\pi}{4}\right) = -2 + \frac{\sqrt{2}}{2}
   $$
   
   To express this as a single fraction:
   $$
   -2 = -\frac{4}{2}
   $$
   $$
   -\frac{4}{2} + \frac{\sqrt{2}}{2} = \frac{-4 + \sqrt{2}}{2}
   $$

**Final Answer:**
$$
\boxed{\dfrac{-4+\sqrt{2}}{2}}
$$
The exact value of the expression is $$ \frac{-4 + \sqrt{2}}{2} $$.

Find the exact value of the expression without using a calculator.

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