Answer:
q = 4.9
m∠R = 27.1°
m∠S = 136.9°
Step-by-step explanation:
To solve the triangle we will use the cosine rule and the sine rule
In the given triangle
∵ q is the opposite side of ∠Q
∵ s is the opposite side of ∠S
∵ r is the opposite side of ∠R
∵ m∠Q = 16°
∵ r = 8 and s = 12
→ By using the cosine rule to find q
∴ q² = r² + s² - 2 r s cos∠Q
→ Substitute the values of r, s, and ∠Q in the rule above
∵ q² = (8)² + (12)² - 2(8)(12)cos(16°)
∴ q² = 23.43775438
→ Take √ for both sides
∴ q = 4.841255455
→ Round it to the nearest tenth
∴ q = 4.9
→ To find angles R and S we will use the sine rule
∵ [tex]\frac{r}{sin(R)}[/tex] =[tex]\frac{q}{sin(Q)}[/tex] = [tex]\frac{s}{sin(S)}[/tex]
→ Use the first two ratios to find m∠R
∵ [tex]\frac{8}{sin(R)}[/tex] = [tex]\frac{4.841255455}{sin(16)}[/tex]
→ By using cross multiplication
∴ 4.841255455 × sin(R) = 8 × sin(16)
→ Divide both sides by 4.841255455
∴ sin(R) = 0.4554807874
→ Use [tex]sin^{-1}[/tex] to find m∠R
∵ m∠R = [tex]sin^{-1}[/tex] (0.4554807874)
∴ m∠R = 27.0958732
→ Round it to the nearest tenth
∴ m∠R = 27.1°
∵ The sum of the measures of the angle in a triangle is 180°
∴ m∠S + m∠Q + m∠R = 180°
→ Substitute the values of m∠Q and m∠R
∵ m∠S + 16° + 27.1° = 180°
∴ m∠S + 43.1 = 180
→ Subtract 43.1 from both sides
∴ m∠S = 136.9°
