Help! trig/calc problems?

Question

FIND THE EXACT VALUE OF THE EXPRESSION:

1-tan^2 75(deg) + csc^2 15(deg)

&

A boat leaves the entrance of a harbor and travels 76 miles on a bearing of N 65 degrees east. How many miles north and how many miles north and how many miles east from the harbor has the boat traveled?

the answers are either:

A. 32.1 miles north and 68.9 miles east

B. 68.9 north and 32.1 miles east

C. 76 miles north amd 76 miles east

D. 35.4 miles north and 163 miles east

Answer 1

For your expression, just use a calculator. tan means tangent. csc means cosecant, which is 1/sine.

1 - tan^2(75) + csc^2(15)

= 1 - 3.73^2 + 1/(0.26^2)

= 1 - 13.91 + 14.79

= 1.88

For your boat problem, it's a right triangle where the angle is 65 degrees, the hypoteneuse is 76 miles, the adjacent leg is how many miles north, and the opposite leg is how many miles east.

sin(65) = east / 76

east = 68.88 miles

cos(65) = north / 76

north = 32.12 miles

The answer is A.