Answer to Question #232655 in Trigonometry for dizzo

Question

A plane flies from Entebbe towards Kiko for 596 km on a bearing of 232 degrees. It then flies from Kikoto Daro for 880 km on a bearing of 258 degrees. It finally flies to Haraka for 458 km on a bearing of 346 degrees from Daro.

(a) How far north or south of Entebbe is /15mrks

(i) Kiko (ii) Haraka?

(b) How far west or east of Entebbe is Haraka?

(c) Calculate the bearing of Haraka from Entebbe

Answer 1

$$E(0,0)$$

$$K(596\sin(232\degree), 596\cos(232\degree))$$

$$D(596\sin(232\degree)+880\sin(258\degree),$$

$$596\cos(232\degree)+880\cos(258\degree))$$

$$H(596\sin(232\degree)+880\sin(258\degree)+458\sin(346\degree),$$

$$596\cos(232\degree)+880\cos(258\degree)+458\cos(346\degree))$$

(a)

(i)

$$K(596\sin(232\degree), 596\cos(232\degree))$$

$$y=596\cos(232\degree)\ km\approx-367\ km$$

Kiko is 367 km south of Entebbe.

(ii)

$$H(596\sin(232\degree)+880\sin(258\degree)+458\sin(346\degree),$$

$$596\cos(232\degree)+880\cos(258\degree)+458\cos(346\degree))$$

$$y=596\cos(232\degree)+880\cos(258\degree)+458\cos(346\degree))\ km$$

$$\approx-105.5\ km$$

Haraka is 105.5 km south of Entebbe.

(b)

$$H(596\sin(232\degree)+880\sin(258\degree)+458\sin(346\degree),$$

$$596\cos(232\degree)+880\cos(258\degree)+458\cos(346\degree))$$

$$x=596\sin(232\degree)+880\sin(258\degree)+458\sin(346\degree))\ km$$

$$\approx-1441.2\ km$$

Haraka is 1441.2 km west of Entebbe.

(c)

$$H(596\sin(232\degree)+880\sin(258\degree)+458\sin(346\degree),$$

$$596\cos(232\degree)+880\cos(258\degree)+458\cos(346\degree))$$

$$\tan\theta=\dfrac{y}{x}$$

$$=\dfrac{596\cos(232\degree)+880\cos(258\degree)+458\cos(346\degree)}{596\sin(232\degree)+880\sin(258\degree)+458\sin(346\degree)}$$

$$\approx0.0732$$

$$\theta\approx4.1867\degree$$

$$bearing=360\degree-90\degree-\theta\approx265.8\degree$$

The bearing of Haraka from Entebbe is $265.8\degree .$