Full solution

Q. If

\sec q = \frac{25}{7}

, find the value of

\tan q

Understand Relationship: First, we need to understand the relationship between secant and tangent. The secant of an angle in a right triangle is the reciprocal of the cosine, and the tangent is the sine divided by the cosine. Therefore, $\sec q = 1/\cos q$ and $\tan q = \sin q/\cos q$ . We can express $\tan q$ in terms of $\sec q$ as $\tan q = \sin q \cdot \sec q$ .
Find $\cos q$ : Since $\sec q = \frac{25}{7}$ , we can find $\cos q$ by taking the reciprocal of $\sec q$ . $\newline$ Calculation: $\cos q = \frac{1}{\sec q} = \frac{1}{\frac{25}{7}} = \frac{7}{25}$ .
Find $\sin q$ : To find $\sin q$ , we use the Pythagorean identity which states that $(\sin q)^2 + (\cos q)^2 = 1$ . We already know $\cos q = \frac{7}{25}$ , so we can solve for $(\sin q)^2$ . $\newline$ Calculation: $(\sin q)^2 = 1 - (\cos q)^2 = 1 - \left(\frac{7}{25}\right)^2 = 1 - \frac{49}{625} = \frac{625}{625} - \frac{49}{625} = \frac{576}{625}$ .
Calculate $\sin q$ : Now we take the square root of $(\sin q)^2$ to find $\sin q$ . Since we are looking for $\tan q$ , which can be positive or negative depending on the quadrant, we consider the positive square root for $\sin q$ , assuming $q$ is in the first quadrant where both sine and cosine are positive. $\newline$ Calculation: $\sin q = \sqrt{\frac{576}{625}} = \frac{24}{25}$ .
Find $\tan q$ : Finally, we can find $\tan q$ by multiplying $\sin q$ by $\sec q$ . $\newline$ Calculation: $\tan q = \sin q \times \sec q = \left(\frac{24}{25}\right) \times \left(\frac{25}{7}\right) = \frac{24 \times 25}{25 \times 7} = \frac{24}{7}$ .