To determine [tex]\(\tan(\theta)\)[/tex] given that [tex]\(\sec(\theta) = \frac{5}{3}\)[/tex] and [tex]\(\theta\)[/tex] is in Quadrant I, follow these steps:
1. Understand the relationship between secant and cosine:
- The secant function is the reciprocal of the cosine function. Therefore,
[tex]\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\][/tex]
- Given [tex]\(\sec(\theta) = \frac{5}{3}\)[/tex], we can write:
[tex]\[
\frac{1}{\cos(\theta)} = \frac{5}{3}
\][/tex]
2. Solve for [tex]\(\cos(\theta)\)[/tex]:
- Taking the reciprocal of both sides, we get:
[tex]\[
\cos(\theta) = \frac{3}{5}
\][/tex]
3. Use the Pythagorean identity to find [tex]\(\sin(\theta)\)[/tex]:
- The Pythagorean identity states:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
- Substituting [tex]\(\cos(\theta) = \frac{3}{5}\)[/tex] into the identity, we get:
[tex]\[
\sin^2(\theta) + \left(\frac{3}{5}\right)^2 = 1
\][/tex]
- Simplify the equation:
[tex]\[
\sin^2(\theta) + \frac{9}{25} = 1
\][/tex]
- Subtract [tex]\(\frac{9}{25}\)[/tex] from both sides:
[tex]\[
\sin^2(\theta) = 1 - \frac{9}{25}
\][/tex]
- Convert [tex]\(1\)[/tex] to a fraction with the same denominator:
[tex]\[
\sin^2(\theta) = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
\][/tex]
- Taking the square root of both sides and considering that [tex]\(\theta\)[/tex] is in Quadrant I (where sine is positive), we get:
[tex]\[
\sin(\theta) = \sqrt{\frac{16}{25}} = \frac{4}{5}
\][/tex]
4. Calculate the tangent function:
- The tangent function is defined as the ratio of the sine function to the cosine function:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
- Substitute the values of [tex]\(\sin(\theta)\)[/tex] and [tex]\(\cos(\theta)\)[/tex]:
[tex]\[
\tan(\theta) = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{5} \div \frac{3}{5}
\][/tex]
- Simplify the division:
[tex]\[
\tan(\theta) = \frac{4}{5} \times \frac{5}{3} = \frac{4 \times 5}{5 \times 3} = \frac{4}{3}
\][/tex]
Therefore, the exact value of [tex]\(\tan(\theta)\)[/tex] is:
[tex]\[
\tan(\theta) = \frac{4}{3}
\][/tex]
