Given triangle DEF find side d and angle E. Round to tenths! angle D is 65 degrees. side DF is 6 and side DE is 9.

d is 8.4 units

m∠E is 40.3°

Step-by-step explanation:

Let us use the cosine rule to find the side d and then use the sine rule to find the measure of angle E

In Δ DEF

∵ EF is represented by d

∵ DF = 6 units and is represented by e

∴ e = 6

∵ DE = 9 units and is represented by f

∴ f = 9

∵ m∠D = 65°

- Substitute the values of e, f and m∠D in the cosine rule above

∴ d² = (6)² + (9)² - 2(6)(9)(cos 65°)

∴ d² = 71.35722773

- Take √  for both sides

∴ d = 8.447320743

- Round it to the nearest tenth

∴ d = 8.4 units

Now let us use the sine rule to find m∠E

∵

- By using cross multiplication

∴ 8.4 × sin(E) = 6 × sin(65)

∴ 8.4 sin(E) = 6 sin(65)

- Divide both sides by 8.4

∴ sin(E) = 0.647362705

- Use to find m∠E

∴ E = (0.647362705)

∴ m∠E = 40.34305

- Round it to the nearest tenth

∴ m∠E = 40.3°