13-6 Success for English Language Learners

LESSON Success for English Language Learners 13-6 The Law of Cosines Steps for Success Step I In order to create interest in the lesson opener, point out the following to students. • Trapeze, trapezium (a geometric figure as well as a bone in the wrist), trapezoid (also a geometric figure and a bone in the wrist), and trapezius (a muscle in the back) all come from the same root. They come from the Greek word trapeza, or “four-legged table.” Step II Teach the lesson. • Have students derive the remaining formulas in the Law of Cosines that were not derived in the text. • Technically speaking, a knot is not a nautical mile. A knot is a speed of one nautical mile per hour. • Heron’s Formula is also known as Hero’s Formula. A Heronian triangle is a triangle having rational side lengths and a rational area. Step III Ask English Language Learners to complete the worksheet for this lesson. • Point out that Example 1A in the student textbook is supported by Problem 1 on the worksheet. Remind students that, for example, side a is opposite angle A, not adjacent to it. • Point out that Example 3 in the student textbook is supported by Problem 2 on the worksheet. • Think and Discuss supports the problems on the worksheet.

Making Connections

• Students comfortable with matrices may wish to verify the following equation: If s s a s b s c is the area of the triangle under consideration, then

2 1 11 a 4 2 a 2 b 2 c 2 1 1 1 b 2 . 2 111 c

LESSON Success for English Language Learners 13-6 The Law of Cosines Problem 1 Use the given measurements to solve ABC.

B 100

b 2 a 2 c 2 2ac cos B c 5 a 7 b 2 7 2 5 2 2 7 5cos 100

A C b

Problem 2 Find the measure of the largest angle, B.

3 4 2 3 0 2 2 2 2 2 30 22 cos B Substitute.

1156 900 484 1320cos B Evaluate exponents. 1156 1384 1320cos B Combine like terms. 228 1320cos B

Subtract 1384 from both sides. 0.1727 cos B

Divide both sides by 1320. mB 80.1

Take Cos 1 of both sides.

Think and Discuss 1. Explain how you use the Triangle Sum Theorem in Example 1 to find the third angle measure.

______2. In Heron’s Formula, what does the quantity s represent?

______

Lesson 13-6 Lesson 14-5 1. Since the sum of the angles in a triangle 1. Modify the more complicated side so that is 180, I can add the measures of the it matches the simpler side. two known angles and subtract the sum 2. Use an identity that allows you to work from 180. That gives me the measure of with like terms to simplify the expression. the third angle. 3. Rewrite the expression so that it is __ . 2. s is half of the perimeter of the triangle. 2 Lesson 14-6 CHAPTER 14 1. You can use algebra or you can use a Lesson 14-1 graph. 1. A function is periodic if it repeats its 2. You can combine like terms and multiply pattern exactly. and divide each side by the same number. You can also factor the equation. 2. The amplitude is the number before the sine or cosine. 3. Use the Quadratic Formula if the equation cannot be factored easily. 3. The period is the number multiplied by the x in a sine or cosine function.

Lesson 14-2 1. The period is __ . b 2. Use the period to find the first x-intercept and then add the period to find the next x-intercept. n 3. The asymptotes are located at x ___ , b where n is an integer.

Lesson 14-3 1. By justifying each step as you simplify, you can prove that both sides of the trigonometric identity are equal. 2. From the Pythagorean Identity, you know that si n 2 is the same as 1 co s 2 . 3. From the Reciprocal Identities, you know that sec ____1 . cos Lesson 14-4 1. You can write sin 75 as the sum of sin 30 and sin 45, which are known values. 2. You need to justify the steps to make the sides of the identity match. 3. Modify the more complicated side until it matches the simpler side.