College Physics by Openstax Chapter 3 Problem 20: Solving Triangle Side and Angle Using Vector and Trigonometric Methods

Problem:

A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors A from B in Figure 3.59. She then correctly calculates the length and orientation of the third side C. What is her result?

Solution:

Consider the illustration shown.

We need to solve for an interior angle of the triangle. So, we need to solve for the value of α first. This can be done by simply subtracting the sum of 21 and 11 degrees from 90 degrees.

\begin{align*} \alpha & = 90 ^ \circ -\left( 21^\circ +11^\circ \right) \\ \alpha & = 58^\circ \end{align*}

Then, using the cosine law, we can now solve for the magnitude of vector C. That is

\begin{align*} C^2 & = A^2 + B^2 - 2AB \cos \alpha \\ C^2 & = \left( 80\ \text{m} \right)^2+\left( 105\ \text{m} \right)^2-2\left( 80\ \text{m} \right)\left( 105\ \text{m} \right) \cos 58^\circ \\ C^2 & = 8522.3564 \\ C & = \sqrt{8522.3564} \\ C & = 92.3 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Before we can solve for the value of θ, we need to know the value of β first. This can be done by using the sine law.

\begin{align*} \frac{\sin \beta}{80\ \text{m}} & = \frac{\sin 58^\circ }{92.3 \ \text{m}} \\ \sin \beta & = \frac{\left( 80 \ \text{m} \right)\sin 58^\circ }{92.3 \ \text{m}} \\ \beta & = \arcsin \left[ \frac{\left( 80 \ \text{m} \right)\sin 58^\circ }{92.3 \ \text{m}} \right] \\ \beta & = 47.3^\circ \end{align*}

Finally, we can solve for θ.

\begin{align*} \theta & = \left( 90 ^\circ +11^\circ \right) - \beta \\ \theta & = \left( 90 ^\circ +11^\circ \right) - 47.3^\circ \\ \theta & = 53.7^\circ \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

College Physics Chapter 3 Problems

Problem 1	Problem 2	Problem 3	Problem 4	Problem 5
Problem 6	Problem 7	Problem 8	Problem 9	Problem 10
Problem 11	Problem 12	Problem 13	Problem 14	Problem 15
Problem 16	Problem 17	Problem 18	Problem 19	Problem 20
Problem 21	Problem 22	Problem 23	Problem 24	Problem 25
Problem 26	Problem 27	Problem 28	Problem 29	Problem 30
Problem 31	Problem 32	Problem 33	Problem 34	Problem 35
Problem 36	Problem 37	Problem 38	Problem 39	Problem 40
Problem 41	Problem 42	Problem 43	Problem 44	Problem 45
Problem 46	Problem 47	Problem 48	Problem 49	Problem 50
Problem 51	Problem 52	Problem 53	Problem 54	Problem 55
Problem 56	Problem 57	Problem 58	Problem 59	Problem 60
Problem 61	Problem 62	Problem 63	Problem 64	Problem 65
Problem 66	Problem 67	Problem 68	Problem 69	Problem 70