Question

find the sine, cosine, and tangent of ∠q.
triangle with vertices s, r, q; right angle at r; rs = 30, rq = 16
simplify your answers and write them as proper fractions, improper fractions, or whole numbers.
sin(q) =
cos(q) =
tan(q) =

Explanation:

Step1: Find the hypotenuse SQ

In right triangle \( \triangle RQS \), by Pythagorean theorem \( SQ=\sqrt{RS^{2}+RQ^{2}} \). Given \( RS = 30 \), \( RQ = 16 \), so \( SQ=\sqrt{30^{2}+16^{2}}=\sqrt{900 + 256}=\sqrt{1156}=34 \).

Step2: Calculate \( \sin(Q) \)

\( \sin(Q)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{RS}{SQ}=\frac{30}{34}=\frac{15}{17} \).

Step3: Calculate \( \cos(Q) \)

\( \cos(Q)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{RQ}{SQ}=\frac{16}{34}=\frac{8}{17} \).

Step4: Calculate \( \tan(Q) \)

\( \tan(Q)=\frac{\text{opposite}}{\text{adjacent}}=\frac{RS}{RQ}=\frac{30}{16}=\frac{15}{8} \).

Answer:

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