Question

find m∠t.

write your answer as an integer or as a decimal rounded to the nearest tenth.

m∠t = °

Explanation:

Step1: Identify triangle type and sides

Triangle \( STU \) is right - angled at \( S \). So, \( \angle S = 90^{\circ} \), \( SU = 3.1 \) (adjacent to \( \angle T \)), and \( TU = 3.6 \) (hypotenuse). We can use the cosine function to find \( \angle T \) since \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \).

Step2: Apply cosine formula

For \( \angle T \), \( \cos(\angle T)=\frac{ST}{TU} \)? Wait, no. Wait, in right - triangle \( STU \), with right angle at \( S \), the side adjacent to \( \angle T \) is \( ST \)? Wait, no, wait. Wait, \( SU = 3.1 \), \( TU = 3.6 \), and \( \angle S = 90^{\circ} \). So, the side adjacent to \( \angle T \) is \( ST \)? Wait, no, let's label the sides correctly. In right - triangle \( STU \), right - angled at \( S \), the sides are: \( SU \) (one leg), \( ST \) (another leg), and \( TU \) (hypotenuse). We know \( SU = 3.1 \), \( TU = 3.6 \). To find \( \angle T \), we can use the cosine of \( \angle T \), where \( \cos(\angle T)=\frac{\text{adjacent side to } \angle T}{\text{hypotenuse}} \). The adjacent side to \( \angle T \) is \( ST \)? Wait, no, wait. Wait, \( \angle T \) is at vertex \( T \), so the sides: the side opposite to \( \angle T \) is \( SU \), and the adjacent side is \( ST \), and hypotenuse is \( TU \). Wait, we know \( SU = 3.1 \) (opposite to \( \angle T \)) and \( TU = 3.6 \) (hypotenuse). So we can use the sine function: \( \sin(\angle T)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{SU}{TU}=\frac{3.1}{3.6} \).
Let's calculate \( \frac{3.1}{3.6}\approx0.8611 \). Then \( \angle T=\sin^{- 1}(0.8611) \).
Using a calculator, \( \sin^{-1}(0.8611)\approx59.4^{\circ} \) (rounded to the nearest tenth).
Wait, alternatively, if we consider the adjacent side. Wait, maybe I made a mistake. Let's re - examine. In right - triangle \( STU \), right - angled at \( S \), so \( \angle S = 90^{\circ} \), \( TU \) is hypotenuse (\( c = 3.6 \)), \( SU \) is one leg (\( a = 3.1 \)), \( ST \) is the other leg (\( b \)). To find \( \angle T \), the adjacent side to \( \angle T \) is \( ST \), and the opposite side is \( SU \). But we can also use the cosine of \( \angle U \), but no, we need \( \angle T \). Wait, let's use the cosine of \( \angle T \): \( \cos(\angle T)=\frac{ST}{TU} \), but we don't know \( ST \). We can find \( ST \) using Pythagoras: \( ST=\sqrt{TU^{2}-SU^{2}}=\sqrt{3.6^{2}-3.1^{2}}=\sqrt{12.96 - 9.61}=\sqrt{3.35}\approx1.83 \). Then \( \cos(\angle T)=\frac{ST}{TU}=\frac{1.83}{3.6}\approx0.5083 \), then \( \angle T=\cos^{-1}(0.5083)\approx59.5^{\circ} \). Wait, but when we use sine: \( \sin(\angle T)=\frac{SU}{TU}=\frac{3.1}{3.6}\approx0.8611 \), \( \sin^{-1}(0.8611)\approx59.4^{\circ} \). The difference is due to rounding. Let's use the sine function more accurately. \( \frac{3.1}{3.6}=31\div36\approx0.861111 \). Then \( \sin^{-1}(0.861111) \). Using a calculator, \( \sin^{-1}(0.8611)\approx59.4^{\circ} \) (rounded to the nearest tenth).

Answer:

\( 59.4 \) (The answer may vary slightly depending on the calculator's precision, but approximately \( 59.4^{\circ} \) when rounded to the nearest tenth)