Calculating Angle C in Vertical Plane: Trigonometric Problem in Dynamics Book

In summary, The conversation is about calculating the angle C in a given diagram and using various hints and formulas to find the solution. The conversation also discusses finding the height y and the magnitude of the velocity at point C. The height is determined using the lengths of AC and BC, while the velocity is decomposed into two components and calculated using the angular velocity measured at A. Ultimately, the conversation is successful in finding the solution.
  • #1
Pyrrhus
Homework Helper
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Hello, I'm in need of a hint or few pointers on how to calculate the angle C of the picture attached. I've already calculated y.

I was doing a few problems in this Dynamics book, i bought recently, and the ascention angle (angle C) is beating me :eek:

"The airplane C is being tracked down by the radar stations A and B. At the instant shown on the picture, the triangle ABC encounters itself in vertical plane and the lectures are Angle A = 30 degrees, Angle B = 22 degrees, Angular Speed A = 0.026 rad/s, Angular Speed B = 0.032 rad/s. Find a) the height y, b) the magnitude of the velocity (the vector V is at point C directed at an ascention angle (angle C) with respect tot he horizontal c) the ascention angle at the instant shown (angle c)"

ah yes distance d = 1000 m and it's between the stations A and B.

I hope the diagram is clear enough...
 

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  • #2
A few hints:
1) Having "y", it is easy determine lengths of AC, BC, and form the vector from A to C, and the vector from B to C.
Let for example the vector from AC have the form [tex]\vec{r}_{AC}=r_{AC}\hat{r}_{AC}[/tex]
where [tex]r_{AC},\hat{r}_{AC}[/tex] are the length and direction vector, respectively.

2) Let [tex]\hat{n}_{AC}[/tex] be the unit vector in the plane of the triangle perpendicular to [tex]\hat{r}_{AC}[/tex] and pointing in the direction of increasing angle, and make a similar construction for [tex]\hat{n}_{BC}[/tex]

3) Decompose your velocity as:
[tex]\vec{v}=v_{AC}\hat{r}_{AC}+v_{BC}\hat{r}_{BC}[/tex]

4) We therefore have, for example the equality:
[tex]\vec{v}\cdot\hat{n}_{AC}=r_{AC}\omega_{AC}\to{v}_{BC}\hat{r}_{BC}\cdot\hat{n}_{AC}}=r_{AC}\omega_{AC}\to{v}_{BC}=\frac{r_{AC}\omega_{AC}}{\hat{r}_{BC}\cdot\hat{n}_{AC}}[/tex]
where [tex]\omega_{AC}[/tex] is the angular velocity measured at A.

5) Thus, we have determined [tex]\vec{v}[/tex] and may answer the two remaining questions.
Remember that [tex]\hat{r}_{AC},\hat{r}_{BC}[/tex] are not orthogonal vectors!
 
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  • #3
Thanks Arildno, i was able to solve it. :biggrin:
 

1. What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. These functions represent the ratios between the sides of a right triangle and can be used to solve for missing angles or sides in a triangle.

2. What is the unit circle and how is it related to trigonometry?

The unit circle is a circle with a radius of 1 centered at the origin of a Cartesian coordinate system. It is used in trigonometry to relate the values of the trigonometric functions to the coordinates of points on the circle. This allows for the extension of trigonometric functions to any angle, not just those in a right triangle.

3. How do I solve trigonometric problems using the Pythagorean identity?

The Pythagorean identity, which states that sin²θ + cos²θ = 1, is often used to simplify trigonometric expressions and solve equations involving trigonometric functions. By rearranging this identity, you can solve for one trigonometric function in terms of the others, making it a useful tool in solving trigonometric problems.

4. What is the difference between radians and degrees in trigonometry?

Radians and degrees are two units of measurement for angles. Radians are a unit of measurement based on the radius of a circle, where 2π radians is equivalent to a full circle. Degrees, on the other hand, are a unit of measurement based on dividing a circle into 360 equal parts. Radians are often used in trigonometry because they simplify calculations involving trigonometric functions.

5. How can I use trigonometry to solve real-world problems?

Trigonometry has many real-world applications, such as in architecture, engineering, and navigation. By using trigonometric functions, you can solve problems involving right triangles, such as finding distances or heights of objects, determining angles between objects, and calculating forces and vectors. Trigonometry is also useful in analyzing periodic phenomena, such as sound waves and electromagnetic waves.

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