Question

if \\( \tan ( \theta ) = \frac { 8 } { 6 } \\), \\( 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

Given \(\tan(\theta)=\frac{8}{6}=\frac{4}{3}\). We know that \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). Let the opposite side \(a = 8\), adjacent side \(b=6\). By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(c\) is the hypotenuse.

$$c=\sqrt{8^{2}+6^{2}}=\sqrt{64 + 36}=\sqrt{100}=10$$

Step2: Calculate \(\sin(\theta)\)

\(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\)

$$ \sin(\theta)=\frac{8}{10}=\frac{4}{5}$$

Step3: Calculate \(\cos(\theta)\)

\(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\)

$$ \cos(\theta)=\frac{6}{10}=\frac{3}{5}$$

Step4: Calculate \(\sec(\theta)\)

Since \(\sec(\theta)=\frac{1}{\cos(\theta)}\)

$$ \sec(\theta)=\frac{1}{\frac{3}{5}}=\frac{5}{3}$$

Answer:

\(\sin(\theta)=\frac{4}{5}\), \(\cos(\theta)=\frac{3}{5}\), \(\sec(\theta)=\frac{5}{3}\)