It's Time To Solve Trigonometry Questions

Explanation
When you are given 2 sides and an included angle of a triangle, you can apply the cosine rule. The cosine rule is used to find the length of the third side of a triangle when the lengths of the other two sides and the included angle are known. It states that the square of the third side is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle. Therefore, cosine is the appropriate rule to use in this scenario.

Explanation
This case will not give rise to an ambiguity when using the Sine rule because the given values of C, a, and c are sufficient to determine a unique solution. The Sine rule states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In this case, we have the values of angle C, side a, and side c, which allows us to calculate the value of angle A using the Sine rule. Therefore, there is no ambiguity in this case.

Explanation
Yes, this case will give rise to an ambiguity when using the Sine rule. The Sine rule states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant. However, in this case, we are given the angle A as 680 degrees, which is not possible in a triangle. Therefore, it is not possible to apply the Sine rule accurately in this scenario, leading to ambiguity.

Explanation
To calculate angle ACB, we can use the Law of Cosines. According to this law, in a triangle with sides a, b, and c and angle C opposite side c, the formula is c^2 = a^2 + b^2 - 2ab*cos(C). Plugging in the given values, we get 47^2 = 31^2 + 53^2 - 2*31*53*cos(C). Solving this equation gives us cos(C) = (31^2 + 53^2 - 47^2) / (2*31*53). Taking the inverse cosine of this value, we find that angle ACB is approximately 35.5 degrees.

Explanation
To find angle CAB, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds true: c^2 = a^2 + b^2 - 2ab*cos(C). In this case, we have side AC = 53, side AB = 74, and angle ACB = 68 degrees. Plugging these values into the equation, we get 74^2 = 53^2 + 74^2 - 2*53*74*cos(68). Solving for cos(68), we find that cos(68) = (53^2 + 74^2 - 74^2) / (2*53*74). Taking the inverse cosine of this value, we find that angle CAB is approximately 70.4 degrees.

Explanation
The shortest distance from point C to line AB can be found by constructing a perpendicular line from point C to line AB. This perpendicular line will intersect line AB at a right angle. Using the given values, we can use the Pythagorean theorem to find the length of the perpendicular line. The lengths of the sides of the right triangle formed are 25 (AC), 28 (BC), and the unknown length of the perpendicular line. By solving for the unknown length using the Pythagorean theorem, we find that it is equal to 14. Therefore, the shortest distance from C to AB is 14 units.

Explanation
The height of the mast can be found using trigonometric ratios. Since C is the foot of the mast and A is the point from where the angle of elevation is measured, we can form a right triangle ACT. The angle of elevation of the top of the mast is given as 40 degrees. Using the tangent function, we can write the equation tan(40) = height of mast / CT. Solving for the height of the mast, we get height of mast = tan(40) * CT. Since the value of CT is not given, we cannot calculate the exact height. Therefore, the answer is given to the nearest integer, which is 21.