Question

question 21
find the exact value of the expression.
\\( \sec \left( \frac { \pi } { 4 } \
ight) = \\)
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question 22
find the exact value of the expression.
\\( \tan \left( \frac { \pi } { 3 } \
ight) = \\)
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Explanation:

Step1: Recall the definition of secant

The secant function is defined as $\sec(x)=\frac{1}{\cos(x)}$. So for $x = \frac{\pi}{4}$, we have $\sec(\frac{\pi}{4})=\frac{1}{\cos(\frac{\pi}{4})}$.

Step2: Find the value of $\cos(\frac{\pi}{4})$

We know that $\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$.

Step3: Calculate $\sec(\frac{\pi}{4})$

Substitute $\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$ into the formula $\sec(\frac{\pi}{4})=\frac{1}{\cos(\frac{\pi}{4})}$. Then $\sec(\frac{\pi}{4})=\frac{1}{\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$.

Step4: Recall the definition of tangent

The tangent function is defined as $\tan(x)=\frac{\sin(x)}{\cos(x)}$. For $x=\frac{\pi}{3}$, we have $\tan(\frac{\pi}{3})=\frac{\sin(\frac{\pi}{3})}{\cos(\frac{\pi}{3})}$.

Step5: Find the values of $\sin(\frac{\pi}{3})$ and $\cos(\frac{\pi}{3})$

We know that $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ and $\cos(\frac{\pi}{3})=\frac{1}{2}$.

Step6: Calculate $\tan(\frac{\pi}{3})$

Substitute $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ and $\cos(\frac{\pi}{3})=\frac{1}{2}$ into the formula $\tan(\frac{\pi}{3})=\frac{\sin(\frac{\pi}{3})}{\cos(\frac{\pi}{3})}$. Then $\tan(\frac{\pi}{3})=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$.

Answer:

$\sec(\frac{\pi}{4})=\sqrt{2}$; $\tan(\frac{\pi}{3})=\sqrt{3}$