Question

if $\theta=\frac{\pi}{4}$, then find exact values for the following:
$\sec(\theta)$ equals
$\csc(\theta)$ equals
$\tan(\theta)$ equals
$\cot(\theta)$ equals
question help: video

Explanation:

Step1: Recall trigonometric identities

We know that \(\sec\theta=\frac{1}{\cos\theta}\), \(\csc\theta = \frac{1}{\sin\theta}\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and \(\cot\theta=\frac{\cos\theta}{\sin\theta}\).
When \(\theta=\frac{\pi}{4}\), \(\sin(\frac{\pi}{4})=\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\).

Step2: Calculate \(\sec(\theta)\)

Substitute \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\) into \(\sec\theta=\frac{1}{\cos\theta}\).
\(\sec(\frac{\pi}{4})=\frac{1}{\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}\).

Step3: Calculate \(\csc(\theta)\)

Substitute \(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\) into \(\csc\theta=\frac{1}{\sin\theta}\).
\(\csc(\frac{\pi}{4})=\frac{1}{\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}\).

Step4: Calculate \(\tan(\theta)\)

Substitute \(\sin(\frac{\pi}{4})=\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\) into \(\tan\theta=\frac{\sin\theta}{\cos\theta}\).
\(\tan(\frac{\pi}{4})=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\).

Step5: Calculate \(\cot(\theta)\)

Substitute \(\sin(\frac{\pi}{4})=\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\) into \(\cot\theta=\frac{\cos\theta}{\sin\theta}\).
\(\cot(\frac{\pi}{4})=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1\).

Answer:

\(\sec(\frac{\pi}{4})=\sqrt{2}\), \(\csc(\frac{\pi}{4})=\sqrt{2}\), \(\tan(\frac{\pi}{4}) = 1\), \(\cot(\frac{\pi}{4})=1\)