Question

the terminal point ( p(x,y) ) determined by a real number ( t ) is given. find ( sin t ), ( cos t ), and ( \tan t ).
( left( \frac { 5 } { 13 }, \frac { 12 } { 13 }
ight) )
( sin t = )
( cos t = )
( \tan t = )

Explanation:

Step1: Recall the definitions

For a terminal point \(P(x,y)\) on the unit circle (here \(x = \frac{5}{13}\), \(y=\frac{12}{13}\)), \(\sin t=y\), \(\cos t = x\), and \(\tan t=\frac{y}{x}\).

Step2: Calculate \(\sin t\)

Since \(y = \frac{12}{13}\), \(\sin t=\frac{12}{13}\).

Step3: Calculate \(\cos t\)

Since \(x=\frac{5}{13}\), \(\cos t=\frac{5}{13}\).

Step4: Calculate \(\tan t\)

\(\tan t=\frac{y}{x}=\frac{\frac{12}{13}}{\frac{5}{13}}=\frac{12}{5}\).

Answer:

\(\sin t=\frac{12}{13}\), \(\cos t=\frac{5}{13}\), \(\tan t=\frac{12}{5}\)