Math 2HonorsName______

Lesson 6-5: Law of Cosines

Learning Goals:

  • I can draw an altitude to create two right triangles and can establish the relationships of the sides in each right triangle using the sine and cosine functions of a single angle in the original triangle.
  • I can derive the Law of Cosines using the Pythagorean Theorem, two right triangles forms by drawing an altitude, and substitution.
  • I can generalize the Law of Cosines to apply to each included angle.
  • I can use the Law of Cosines to solve real world problems.
  1. Derivations of the Law of Cosines differ depending on whether the angle under consideration is acute or obtuse. In below, is an obtuse angle and h is the length of the altitude from vertex C.
  1. Explain why each step in the following derivation is correct. The reason for Step 5 is that as suggested by the figure in problem #5 part c of Lesson 5-1 Practice.

(1)

(2)

(3)

(4)

(5)

(6)

  1. For the case of an acute angle, draw a new triangle ABC with acute, and draw the altitude from vertex C to side Length c will then be divided into two parts of lengths x and c – x. The rest of the derivation of the Law of Cosines is similar to the obtuse case. Try to construct the argument.
  1. The Law of Cosines states a relationship among the lengths of three sides of a triangle and the cosine of an angle of the triangle. If you know the lengths of two sides and the measure of the angle between the two sides, you can use the Law of Cosines to calculate the length of the third side.
  1. Writes the Law of Cosines to calculate the length a in if you know the lengths b and c and the measure of
  1. Write a third form of the Law of Cosines for for when you know the measure of
  1. Suppose in you needed to calculate the length of What information would you need in order to use the Law of Cosines? Write the equation that you would use.
  1. Suppose in you know and lengths p and r.
  1. Could you find the length q using the Law of Cosines? Explain your reasoning.
  1. Write an equation for calculating the length q.
  1. Using the information in Part d, write an expression for finding the length q using the Law of Sines. Which method would you prefer to use? Why?
  1. Surveyors are often faced with irregular polygonal regions for which they are asked to locate and stake out boundaries, determine elevations, and estimate areas. Some of these tasks can be accomplished by using a site map and a transit as shown in the photo below. In one subdivision of property near a midsize city, a plot of land had the shape and dimensions shown.

Examine the triangulation of the plot shown below.

  1. Find AC to the nearest thousandth.
  1. Find AD to the nearest thousandth.
  1. The Law of Cosines, states a relation among the lengths of three sides of a triangle ABC and the cosine of an angle of the triangle. If you know the lengths of all three sides of the triangle, you can calculate the cosine of any angle and then determine the measure of the angle itself.
  1. Solve the equation for
  1. Using your results from Problem 3, find the measure of to the nearest thousandth of a degree. What is the measure of the third angle in
  1. Find the area of
  1. Explain how you could determine the area of the entire pentagonal plot.
  1. As you have seen, the Law of Sines and Law of Cosines can be used to find the measures of unknown angles as well as unknown sides of any triangle. You need to study given information about side and angle measurements to decide which law to apply. Then you work with the resulting equations to solve for the unknown angle or side measurements.

For example, suppose that two sides, and and a diagonal, of a parallelogram ABCD measure 7 cm, 9 cm, and 11 cm, respectively.

  1. Draw and label a sketch of parallelogram ABCD.
  1. Which of the two trigonometric laws can be used to find the measure of an angle in that parallelogram?
  1. Find the measure of that angle to the nearest thousandth of a degree.
  1. The diagonal splits the parallelogram ABCD into two congruent triangles. Find the remaining measures of the angles in those triangles.
  1. Find the length of diagonal