Calculus and Trigonometry: Solving Equations and Triangle Problems

This assignment presents solutions to a variety of trigonometry and triangle problems. It begins by solving trigonometric equations, including finding exact solutions within a specified interval. The assignment then progresses to solving triangles, addressing scenarios where the triangle's sides and angles are provided, and the law of sines and cosines are applied. Additionally, the assignment includes the calculation of a triangle's area using Heron's formula. Finally, a word problem involving bearings is solved, requiring the application of trigonometric principles to determine distances. The solutions are detailed, and the assignment covers a range of concepts related to trigonometry and geometry, including the use of sine rule and cosine rule.

1. Solve, finding all exact solutions in [0, 2π).
a) 3cos(2x)-2sinx^(2)xcos(2x)=0
b) tan(2x + π/6) = -1
Solution 1a: Given .
Now,
Sing zero- factor property we get
Either,
Since,
Since,
Since,
Hence from equation (1), the only solution in are
Solution 1b: Given
Now,
Hence, the only solution in are
2. Solve the triangle(s), if possible. Round each final answer to the nearest hundredth
ß = 30°; b = 60; a = 100. (Side a is opposite angle α. Side b is opposite angle β.)
Solution: Given . By sine rule
3. Side a = 40 ft., side b = 30 ft., side c = 12 ft. (Side a is opposite angle α. Side b is
opposite angle β. Side c is opposite side ϒ.
Solve the triangle. Round each final answer to the nearest hundredth.
Find the area of the triangle.
Solution: Given side . The perimeter

By heron’s formula, the area of triangle is
4. Station A is 3.8 miles south of Station B. The bearing from Station A to a ship is
S42°W. The bearing from Station B to the same ship is S26°W. Find the distance
from Station B to the ship, to the nearest tenth of a mile. Show a diagram of any
triangles you use in your solution, with sides and angles labeled.
Solution:
The angle made at ship is
So by sine rule