Question

consider the triangles.

if $\triangle uvw$ is similar to $\triangle trs$, which equations are true?
select all that apply.

$\square \\ \cos t = \frac{12}{13}$

$\square \\ \sin s = \frac{13}{12}$

$\square \\ \sin t = \frac{5}{13}$

$\square \\ \tan s = \frac{5}{12}$

$\square \\ \tan t = \frac{12}{5}$

Explanation:

Step1: Find the hypotenuse of $\triangle UVW$

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 5\), \(b=12\).
\(c=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Step2: Analyze \(\cos T\)

Since \(\triangle UVW\sim\triangle TRS\), \(\angle T=\angle W\).
\(\cos T=\cos W=\frac{adjacent}{hypotenuse}=\frac{12}{13}\)

Step3: Analyze \(\sin S\)

\(\sin S=\frac{opposite}{hypotenuse}\). The value of \(\sin S\) should be less than \(1\), and \(\frac{13}{12}>1\), so this is false.

Step4: Analyze \(\sin T\)

Since \(\angle T=\angle W\), \(\sin T=\sin W=\frac{5}{13}\)

Step5: Analyze \(\tan S\)

Since \(\angle S=\angle V\), \(\tan S=\tan V=\frac{5}{12}\)

Step6: Analyze \(\tan T\)

Since \(\angle T=\angle W\), \(\tan T=\tan W=\frac{5}{12}
eq\frac{12}{5}\)

Answer:

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