Question

Solve △PQR. Round function values to 4 decimal places. Round final answers to the nearest tenth.

Answer 1

Answer:

The length of q is 7.6 units

The measure of angle R is 31.9°

The measure of angle P is 66.1°

Step-by-step explanation:

Let us use the sine and cosine rules to solve the triangle

In Δ PQR,

p is the opposite side to ∠P

q is the opposite side to ∠Q

r is the opposite side to ∠R

∵ p = 7 units

∵ r = 4 units

∵ m∠Q = 82°

- Angle Q is between p and r so let us find q using cosine rule

∵ q² = p² + r² - 2(p)(r) cos(∠Q)

- Substitute the values of p, r and ∠Q in the rule

∴ q² = (7)² + (4)² - 2(7)(4) cos(82°)

∴ q² = 49 + 16 - 56 cos(82°)

∴ q² = 57.2063

- Take √  for both sides

∴ q = 7.5635

- Round it to the nearest tenth

∴ The length of q is 7.6 units

Now let us use the sine rule to find the measures of ∠R and ∠P

∵ $\frac{q}{sin(Q)}=\frac{r}{sin(R)}=\frac{p}{sin(P)}$

- Let us find sin(R)

∴  $\frac{7.5635}{sin(82)}=\frac{4}{sin(R)}$

- By using cross multiplication

∴ 7.5635 × sin(R) = 4 × sin(82)

∴ 7.5635 sin(R) = 4 sin(82)

- Divide both sides 7.5635

∴ sin(R) = 0.5285

- Use $sin^{-1}$ to find m∠R

∵ m∠R = $sin^{-1}$ (0.5285)

∴ m∠R = 31.9042

- Round it to the nearest tenth

∴ The measure of angle R is 31.9°

∵ The sum of the measures of the interior angles of a Δ is 180°

∴ m∠P + m∠Q + m∠R = 180°

∴ m∠P + 82 + 31.9 = 180

∴ m∠P + 113.9 = 180

- Subtract 113.9 from both sides

∴ m∠P = 66.1°

∴ The measure of angle P is 66.1°