Question

find the sine, cosine, and tangent of ∠s.
diagram: right triangle ust with right angle at u, su = 5, st = 13
simplify your answers and write them as proper fractions, improper fractions, or whole numbers.
sin (s) =
cos (s) =
tan (s) =

Explanation:

Step1: Find the length of UT

In right triangle \( \triangle SUT \), by Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( c = 13 \) (hypotenuse), \( SU = 5 \) (one leg), let \( UT = x \). Then \( 5^2 + x^2 = 13^2 \), so \( x^2 = 13^2 - 5^2 = 169 - 25 = 144 \), so \( x = 12 \). Thus, \( UT = 12 \).

Step2: Calculate \( \sin(S) \)

\( \sin(S) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{UT}{ST} = \frac{12}{13} \)

Step3: Calculate \( \cos(S) \)

\( \cos(S) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{SU}{ST} = \frac{5}{13} \)

Step4: Calculate \( \tan(S) \)

\( \tan(S) = \frac{\text{opposite}}{\text{adjacent}} = \frac{UT}{SU} = \frac{12}{5} \)

Answer:

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