1. Lesson 7

1.6. Explore 2

Mathematics 30-3 Module 5

Module 5: Geometry

 

Thus far you have used the cosine law to determine the length of an unknown side of a triangle. It is also possible to use the cosine law to determine the value of an unknown angle. Try This 2 explores this idea.

 
Try This 2

 

This is a photo of a person looking at a map.

iStockphoto/Thinkstock

Sandra is taking the same orienteering course as Jeremy in Lesson 6. Sandra is standing at Checkpoint A and knows the distances between all the checkpoints, as shown in the diagram. In order to walk in the correct direction, she needs to determine the angle between Checkpoint B and Checkpoint C, represented by θ on the diagram.

 

This shows three checkpoints; A, B, and C. The angle at Checkpoint A is labelled with the theta symbol, the distance between Checkpoint A and Checkpoint C is 33 metres, the distance between Checkpoint A and Checkpoint B is 28 metres, and the distance between Checkpoint B and Checkpoint C is 22 metres.

  1.  
    1. Sketch the diagram. Label the sides and angles appropriately.
    2. Use a “guess and check” strategy with the equation a2 = b2 + c2 − 2bc cos A to determine the value of θ in the diagram.
  2.  
    1. Using a “guess and check” strategy can take a long time. A better strategy may be to solve the formula for A. Try isolating A in the equation a2 = b2 + c2 − 2bc cos A using the following steps. Make sure to write each step on a new line.
      • Subtract b2 and c2 from both sides of the equation.
      • Divide both sides of the equation by −2bc.
      • Apply the inverse cosine (cos−1) to both sides of the equation.
    2. Use the equation you determined in part a. to determine the value of θ in the diagram. How did this value compare to the one you determined in question 1. b?

course folder Save your responses in your course folder.

 

If you know all three sides of a triangle, you can use the cosine law to determine an unknown angle. It is possible to rearrange the equation a2 = b2 + c2 − 2bc cos A to allow you to determine unknown angles more easily.

 

 
This diagram shows a triangle with each side labelled “known” and one angle labelled “unknown.”

 

The equation  can be used to determine cos A. To determine the degree measurement of A, find the inverse cosine (cos−1) of both sides of the equation.

 

 

This shows three checkpoints; A, B, and C. The angle at Checkpoint A is labelled with the theta symbol, the distance between Checkpoint A and Checkpoint C is 33 metres, the distance between Checkpoint A and Checkpoint B is 28 metres, and the distance between Checkpoint B and Checkpoint C is 22 metres.

 

This equation can be used to determine the unknown angle in Try This 2 as follows.

 

Write out the equation.
Substitute the known values.
cos A = 0.7516…

caution

When entering fractions on a calculator, extra care must be taken with brackets. The easiest approach is to always put the numerator in brackets and the denominator in brackets.

 

For this example, you would enter the following in your calculator:

A = cos−1 (0.7516…) Use the inverse cosine (cos−1) to determine A.
A = 41.268…° The rounded value of A is 41.268…°.
A = 41° Round to the nearest degree, unless instructed otherwise.


textbook

Read “Example 2” on pages 262 and 263 of the textbook to see how all three angles of a triangle can be determined if you know all three sides.

 

 

Self-Check 3


textbook

Answer “Build Your Skills” question 4, 6, and 7 from pages 267 to 269. Answer

 

 


glossary
Add cosine law to your copy of Glossary Terms.


formula

Add the following formulas to your copy of Formula Sheet:

  • a2 = b2 + c2 − 2bc cos A
Both strategies should have shown θ to be approximately 41°.
cos−1 cos A = A