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Question

a boat leaves the entrance of a harbor and travels 27 miles on a bearing of north 82 degrees east. how many miles north and how many miles east from the harbor has the boat traveled?

Answer 1

You are given:
hypotenuse = 27 miles
angle = 82° degrees

If you draw a right triangle, you'll see that the direction east is the side adjacent to the angle and the direction north is the side opposite the angle:

From SOH-CAH-TOA we get:

SOH:
sin(angle) = opposite / hypotenuse
sin(82°) = opposite / 27
opposite = 27 * sin(82°)
{distance north} = 26.74 miles north

CAH:
cos(angle) = adjacent / hypotenuse
cos(82°) = adjacent / 27
adjacent = 27 * cos(82°)
{distance east} = 3.76 miles east

Note: Some people like to just memorize the last part equating cosine with x in the unit circle, and sine with y in the unit circle. Then you can jump straight to:
x = 27 cos(82°)
= 3.76 (miles E)

y = 27 sin(82°)
= 26.74 (miles N)

Use whichever method you understand best... once you practice enough should be able to easily use the shortcut.

Answer 2

It depends on the lattitude of the harbour. Since the boat travels on a constant bearing of 082 it is following a rhumb line which is a spiral track and not the shortest distance between two points.

Nevertheless, if the harbour is not too far north or south of the equator we can use plane trig as an approximation. The bearing is given as "north 82 degrees east" which is a bearing of 082 or 8 less than due east.

So it ends up 27 * sin(8) = 3.76 miles north.
27 * cos(8) = 26.7 miles east

Answer 3

27 north 82 east

Answer 4

= 26 Celsius

2/3 + 1/7 = 7(2/3) +3(1/7)

14/21 + 3/21

=17/21

Answer 5

sin(82) = north / 27
= 26.74

cos(82) = east / 27
= 3.76

Answer 6

North = 27sin(8)
East = 27 cos(8)

Answer 7

Ditto Conor M.

Answer 8

23 coconuts.

Answer 9

You need to gen up on trig.