Trigonometry

Let the height of the tower be , and let the foot of the tower be O. The points A, B, C are on a line leading to O. Let the distances of A, B, and C from the foot of the tower be . From the angles of elevation, we have: Since the angles of elevation are increasing (30, 45, 60), the points are getting closer to the tower. So the order is A, B, C, O. The distances between the points are: Now we find the ratio of AB to BC: So, the ratio is .

We need to evaluate where and .

First, convert into a more common inverse trigonometric function like . Let . This means . Cosecant is Hypotenuse/Opposite. So, we have a right-angled triangle with Hypotenuse = 5 and Opposite = 3.

By Pythagoras theorem, Adjacent = . This is a standard 3-4-5 triangle.

From this triangle, we can find .

So, . Now the expression becomes:

We use the sum formula for : So, the original expression is . Since , we have:

We are given two equations: 1. 2.

Step 1: Analyze the first equation We know that (the inverse sine function) gives values in the range for . For the equation , we examine the two cases:

- Case 1: When , . Thus, .

- Case 2: When , . Thus, .

These two cases give us real solutions in the interval . Through solving, we find that there are 2 real solutions for . Hence, .

Step 2: Analyze the second equation Now, we solve . This is a transcendental equation, and we solve it by plotting or numerically approximating. The solutions occur within the interval .

We find 3 real solutions for this equation. Thus, .

Step 3: Calculate Now, we calculate the difference: Thus, the correct answer is .

Let the height of the pillar be . Since the angles and represent the angles subtended by the pillar at points A and B, we can use the tangent of these angles to form two equations. 1. At point A, the angle subtended is , so: This means: where is the horizontal distance from point A to the foot of the pillar. 2. At point B, the angle subtended is , so: This means: where is the horizontal distance from point B to the foot of the pillar. Now, we are given that the length of the line segment AB is 10 meters. So: Substituting the expressions for and from equations (1) and (2): To solve for , find a common denominator: Thus, the height of the pillar is meters.

We are given that: The tangent of an angle is given by: Thus, the given condition becomes: This simplifies to: Therefore: Now, for the maximum value of , the optimal condition occurs when all the angles are equal. Let . Then the condition becomes: Substituting into the expression for : Thus: Therefore, the maximum value of is .

We are given the equations and , and we are tasked with finding the value of .
Step 1: Use the identities for sine and cosine.
We know that: Also, from , we have: Step 2: Analyze the possible solutions.
From both conditions, the only consistent solution is: Thus, the correct answer is:

Step 1: Solve .
We know that: Thus, the general solutions for are: Step 2: Find the range of solutions for in .
The interval given is , so we will determine how many solutions fit into this range by considering the form of the general solution for .
Step 3: Count the solutions.
By evaluating the number of integer values of that satisfy , we find that there are 48 distinct values for .

Let the point of observation on the ground be O. Let the first airplane be at height 4000m above the ground. Let the second airplane be at height meters above the ground. We are given that the angles of elevation of the two airplanes from O are 60° and 30°, respectively. Let the horizontal distance of the first airplane from point O be , and the horizontal distance of the second airplane from point O be . Using the tangent of the angles of elevation, we can set up the following relationships: We know that: Thus: Now, since the first airplane is directly above the second airplane, we have: Equating the values of and , we get: Solving for , we get: Thus, the vertical distance between the two airplanes is: