Download Tension Triangles

Tension Problems Notes
The Tension Triangles that you create will be useful when solving certain
Tension Problems, such as “sign” problems. Use this guide to create your
Tension Triangles and then practice using them on the “Tension Problems
Worksheet” problems 4-6.
1. When solving Tension problems using trigonometry, there are four basic
rules to remember:
a. Find the triangle.
b. Fup = Fdown
i. Consider: When there are two triangles present, what does
that mean about the up and downward forces?
c. Fleft = Fright
d. Use Pythagorean Theorem, sin, cos, tan or the Law of Sine to solve
for the unknowns.
2. Examine your Tension Triangles. These triangles can be used for Tension
problems where you find two triangles present. Notice that there are two
tensions, two upward forces and a left and right force.
3. Choose one color to represent tension and shade in the tension side of
each triangle.
4. Choose a second color to represent the upward forces and shade in the
upward force side of each triangle.
5. Choose a third color to represent the left and right forces and shade in the
left and right force.
6. Cut out your triangles and set them aside. Look at the example problem.
7. Example: Determine forces A and B.
8. Do you have enough information to solve the problem using sin, cos, or
tangent? If not, what options do you have? This is where your Tension
Triangles should come in handy.
9. Set your triangles out so that they look like the triangles in the problem.
A
30°
B
45°
10. Consider sin, cos and tan:
a. Look at the upward force on side A. It is opposite the angle 30° and
you know A is the hypotenuse. Therefore, you can use
sinθ = opp / hyp, where the opposite side is represented by Fup and
the hypotenuse is represented by side A. To solve for the opposite
side, you would have the equation: opp = Asinθ.
b. So solve for the left force, which is adjacent to the angle, you can
use cosθ = adj / hyp, where the adjacent side is represented by Fleft
and the hypotenuse is represented by side A. To solve for the
adjacent side, you would have the equation: adj = Acosθ.
c. But what about the triangle on the right side? For the opposite side,
you have to use the equation opp = Bsinθ and adj = Bcosθ.
d. Whoa, that is a lot of information! Let’s slow down and label our
diagram:
Asinθ
Bsinθ
Acosθ
Bcosθ
11. Now we know how to solve for each side of the triangle, but that still
doesn’t help us solve for force A and B. Well, it actually does. Remember
the rules when solving for tension using trigonometry.
12. After identifying the triangle, remember Fup = Fdown. Since the downward
force in this problem is 15 lbs, then the sum of the two upward forces must
be 15 lbs: Asin30° + Bsin45° = 15.
13. Next, Fleft = Fright. Based on this, it can be stated that Acos30° = Bcos45°.
14. Ok, now that we have all of our information, lets solve:
Acos30° = Bcos45°
A = (Bcos45°) / (cos30°)
A = 0.816B
Now we can substitute this value into our next equation:
Asin30° + Bsin45° = 15
(0.816B) sin30° + Bsin45° = 15
0.408B + 0.707B = 15
1.12B = 15
B = 13.45 lbs
A = 0.816B
A = (0.816) (13.45)
A = 10.98 lbs
15. Not too bad, right? For some of you, this method will actually be quite
easy, but for others, you will prefer the second method.
16. Take a look at your triangle pattern that you set up all the way back in
Step 9. You can rotate these two triangles to form a single triangle as
shown in the diagram below.
Fup
A
B
30°
45°
Fleft
Fup
Fright
30° 45°
A
Fup
Fup
B
17. Use trigonometry to determine the angles in the lower corners of the
triangle.
60°
45°
18. Notice the two bottom components on the bottom of the triangle are the
upward forces. Since the Fup = Fdown, what value is the bottom of the
triangle? If you said 15 lbs, you are correct.
15
19. Now you can use the Law of Sine to solve for A and B.
20. (15 / sin75°) = (A / sin45°)
A = 10.98 lbs
21. (15 / sin75°) = (B / sin60°)
B = 13.45 lbs
22. Now that you have seen both examples on how to solve Tension
Problems, try the two example problems using either sin, cos, tan or the
Law of Sine.