In ΔQRS, r = 9.4 cm, s = 8.8 cm and ∠Q=171°. Find ∠R, to the nearest 10th of a degree.

Given the triangle \( \triangle QRS \) with side lengths \( r = 9.4 \) cm, \( s = 8.8 \) cm, and angle \( \angle Q = 171^\circ \), we need to find \( \angle R \).

To solve this, we can use the Law of Sines, which states:

\[
\frac{\sin \angle Q}{q} = \frac{\sin \angle R}{r} = \frac{\sin \angle S}{s}
\]

However, since we are only given two sides and one angle, we cannot directly apply this law. Instead, we will use the Law of Cosines to find \( q \) first:

\[
q^2 = r^2 + s^2 - 2rs \cdot \cos(\angle Q)
\]

Let's calculate \( q \):

\[
q^2 = 9.4^2 + 8.8^2 - 2(9.4)(8.8) \cdot \cos(171^\circ)
\]

Now, we will calculate \( q \) and then use the Law of Sines to find \( \angle R \).The length of side \( q \) is approximately 18.14 cm.

Next, we use the Law of Sines to find \( \angle R \):

\[
\frac{\sin \angle R}{r} = \frac{\sin \angle Q}{q}
\]

\[
\sin \angle R = \frac{r \cdot \sin \angle Q}{q}
\]

Now, let's calculate \( \sin \angle R \) and then \( \angle R \).The measure of \( \angle R \) is approximately \( 4.6^\circ \) to the nearest tenth of a degree.

Would you like more details on this calculation, or do you have any further questions? 

Here are five related questions you might find interesting:
1. How can we find \( \angle S \) using the information we have?
2. How does the Law of Sines apply to non-obtuse triangles?
3. What would happen if \( \angle Q \) were less than \( 90^\circ \)?
4. How can we verify the accuracy of our calculated angle using another method?
5. How does the Law of Cosines help in solving for angles in obtuse triangles?

**Tip:** When dealing with obtuse triangles, always be mindful of the angles' impact on the cosine and sine values, especially when applying trigonometric laws.

Learn how to calculate angle R in triangle QRS with side lengths r = 9.4 cm, s = 8.8 cm, and angle Q = 171° using the Law of Sines and Law of Cosines. Detailed step-by-step solution provided. Suitable for high school students studying trigonometry.

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