Trigonometric Function Application

In summary, the problem asks to find the sum of the lengths of the perpendiculars dropped from a point in an equilateral triangle to the three sides. It is possible to solve this problem by using the interior point of the triangle as a starting point, but this approach is not always guaranteed to work.
  • #1
nolachrymose
71
0
I was given the following problem to solve (diagram is attached):

In right triangle ACD with right angle at D, B is a point on side AD between A and D. The length of segment AB is 1. Is <DAC = [itex]\alpha[/itex] and <DBC = [itex]\beta[/itex], then find the length of side DC in terms of [itex]\alpha[/itex] and [itex]\beta[/itex].

I have tried all sorts of mixes of the trigonometric functions to solve this problem, but the closest I can get is where all the terms cancel out. Could someone possibly give me a hint as to how to start this problem, as I'm having great difficult solving it? Any help is greatly appreciated. Thank you! :)
 

Attachments

  • trig.gif
    trig.gif
    1.7 KB · Views: 460
Physics news on Phys.org
  • #2
Have you tried law of sines? Think of the set up as two triangles, ABC and ADC. Try to solve first for the one side they have in common (the hypotenuse of ADC).
 
Last edited:
  • #3
nolachrymose said:
I was given the following problem to solve (diagram is attached):



I have tried all sorts of mixes of the trigonometric functions to solve this problem, but the closest I can get is where all the terms cancel out. Could someone possibly give me a hint as to how to start this problem, as I'm having great difficult solving it? Any help is greatly appreciated. Thank you! :)

Yeah, first represent all relevant angles in terms of alpha and beta. Then, use the sin law to solve for BC(in terms of alpha and beta). Then, use simple trig to do the rest.
 
  • #4
While I do know the Law of Sines, I'm not allowed to use it since we haven't proven it yet. However, I was able to solve it by using two simultaneous equations and substitution. My answer was this:

[tex]\frac{\tan{\beta}\tan{\alpha}}{\tan{\beta}-\tan{\alpha}}[/tex]

Thank you for your suggestions, though! This always seems to happen when I post, sorry...

-----------

I have one last question before I can retire this chapter! Unfortunately, it's the hardest one I've come across so far (unless I'm missing something painfully obvious here). It asks to prove that for any point in the interior of an equilateral triangle, the sum of the lengths of the perpendiculars dropped from the point to the three sides is equal to the length of the altitude of the triangle.

I have no idea where to start on this one (which is quite rare). I've tried similar triangles, but the problem is I cannot prove that any of the lines from the interior point to the vertices are angle bisectors or anything, so I end up with triangles that have no relation to the altitude itself.

One possible approach that I could use was to use the interior point of the triangle which is the median, incenter, circumcenter, and orthocenter of the triangle (since it's equilateral), but this too I cannot do without loss of generality.

Any help is greatly appreciated! Thank you! :)
 
Last edited:

FAQ: Trigonometric Function Application

What are some real-world applications of trigonometric functions?

Trigonometric functions have numerous real-world applications, including in architecture, engineering, navigation, physics, and astronomy. For example, architects use trigonometry to calculate the angles and distances of a building's roof, engineers use trigonometry to design bridges and roads, and navigators use trigonometry to determine the position of a ship or aircraft.

How are trigonometric functions used in music?

Trigonometric functions are used in music to describe the wave-like nature of sound. The frequency and amplitude of a sound wave can be represented using trigonometric functions, and this is essential for understanding and creating different musical tones and chords.

Why are trigonometric functions important in calculus?

Trigonometric functions are important in calculus because they are used to describe the relationships between the sides and angles of a right triangle. These relationships, known as trigonometric identities, are essential for solving many types of calculus problems, including those involving derivatives and integrals.

How are trigonometric functions used in computer graphics?

Trigonometric functions are used in computer graphics to create smooth and realistic curves. By using trigonometric equations, computer graphics programs can accurately represent the curves and angles of objects, resulting in more lifelike images and animations.

Can trigonometric functions be used to solve real-life problems?

Yes, trigonometric functions can be used to solve real-life problems. For example, they can be used to determine the height of a building or mountain, the distance between two objects, or the speed and direction of a moving object. Trigonometry is a valuable tool for solving a wide range of practical problems in various fields.

Similar threads

Replies
15
Views
567
Replies
20
Views
843
Replies
5
Views
2K
Replies
5
Views
935
Replies
10
Views
1K
Replies
9
Views
604
Back
Top