Question

find the sine, cosine, and tangent of $\angle t$. simplify your answers and write them as proper fractions, improper fractions, or whole numbers. $\sin(t) = \square$ $\cos(t) = \square$ $\tan(t) = \square$

Explanation:

Step1: Find the hypotenuse \( TS \)

In right triangle \( TUS \), by Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( a = 32 \), \( b = 60 \), \( c = TS \).
\( TS = \sqrt{32^2 + 60^2} = \sqrt{1024 + 3600} = \sqrt{4624} = 68 \)

Step2: Calculate \( \sin(T) \)

\( \sin(T) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{SU}{TS} = \frac{60}{68} = \frac{15}{17} \)

Step3: Calculate \( \cos(T) \)

\( \cos(T) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{TU}{TS} = \frac{32}{68} = \frac{8}{17} \)

Step4: Calculate \( \tan(T) \)

\( \tan(T) = \frac{\text{opposite}}{\text{adjacent}} = \frac{SU}{TU} = \frac{60}{32} = \frac{15}{8} \)

Answer:

\( \sin(T) = \frac{15}{17} \)
\( \cos(T) = \frac{8}{17} \)
\( \tan(T) = \frac{15}{8} \)