In triangle $RST$, which is a right triangle with right angle at $S$, if $R$ is $30°$ and the length of side $RS$ is $10$, what is the length of side $ST$?

• A. $5$
• B. $10$
• C. $20$
• D. $8.66$

Answer: (A)

Verified step by step guidance

1. Identify the given information: triangle RST is a right triangle with the right angle at S, angle R is 30°, and side RS (which is adjacent to angle R) is 10 units long.
2. Recall that in a right triangle, the sides relate to the angles through trigonometric ratios. Since angle R is 30°, and side RS is adjacent to angle R, we can use the cosine function: \(\cos(30^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{RS}{RT}\).
3. Set up the equation using the cosine ratio: \(\cos(30^\circ) = \frac{10}{RT}\). From this, solve for the hypotenuse \(RT\): \(RT = \frac{10}{\cos(30^\circ)}\).
4. Next, to find side ST (which is opposite angle R), use the sine function: \(\sin(30^\circ) = \frac{ST}{RT}\). Substitute \(RT\) from the previous step to express \(ST\) in terms of known values.
5. Finally, solve for \(ST\): \(ST = RT \times \sin(30^\circ)\). Substitute the expression for \(RT\) to get \(ST = \frac{10}{\cos(30^\circ)} \times \sin(30^\circ)\), which you can simplify to find the length of side ST.