Question

if $\tan(\theta)=\frac{24}{7},0\leq\theta\leq\frac{\pi}{2}$, then $\sin(\theta)$ equals $\cos(\theta)$ equals $\sec(\theta)$ equals

Explanation:

Step1: Use the Pythagorean identity for tangent

We know that \(\tan\theta=\frac{24}{7}=\frac{\text{opposite}}{\text{adjacent}}\). By the Pythagorean theorem, the hypotenuse \(r\) of the right - triangle is \(r = \sqrt{24^{2}+7^{2}}\).

$$
LATEXBLOCK0
$$

Step2: Find \(\sin\theta\)

Using the definition \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), we have \(\sin\theta=\frac{24}{25}\) since the opposite side is \(24\) and the hypotenuse is \(25\).

Step3: Find \(\cos\theta\)

Using the definition \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), we have \(\cos\theta=\frac{7}{25}\) since the adjacent side is \(7\) and the hypotenuse is \(25\).

Step4: Find \(\sec\theta\)

Using the reciprocal identity \(\sec\theta=\frac{1}{\cos\theta}\), we substitute \(\cos\theta = \frac{7}{25}\). So \(\sec\theta=\frac{25}{7}\).

Answer:

\(\sin(\theta)=\frac{24}{25}\), \(\cos(\theta)=\frac{7}{25}\), \(\sec(\theta)=\frac{25}{7}\)