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Learning Objective We will determine1 how to find the side lengths of right triangles by using the Sine and Cosine Ratio2. What are we going to do? What is Ratio means?_______. CFU Activate Prior Knowledge The Greek letter theta ( ) is used to represent the measure of an angle in a right triangle. Opposite(Opp) – Is a position on the other side of a specific angle from; facing. Hypotenuse Opposite Adjacent(Adj) -next to something else. Hypotenuse(Hyp )- the longest side of a right triangle, opposite the right angle (90 ). o Adjacent On your white board, write the Hypotenuse side of: Angle A Angle B CFU B C A Students, you already know identify the sides in a right triangle. Today, we will learn how to find the side lengths of a right triangles by using the Sine and Cosine Ratios Ratio. Make Connection On your white board, write the opposite side of: Angle A Angle B CFU B C A On your white board, write the Adjacent side of: Angle A Angle B CFU B C A AB 1 Figure out 2 A ratio is a comparison between two different things, written as a fraction. Vocabulary AB

Concept Development A trigonometric ratio is a ratio of two sides of a right triangle. You have already seen one trigonometric ratio, the tangent. There are two additional trigonometric ratios, the sine and the cosine, that involve the hypotenuse of a right triangle On your whiteboard, Identify the following: What is the ratio for Sin (T)? What is the ratio for Sin (G)? In your own words, describe the sine. In Trigonometry, Sine is_____. CFU In a right triangle, the sine of an acute angle is the ratio of the length of the opposite leg over the length of the hypotenuse. Sin (T) = 8/17 G T R 17 8 15 Sin (G) = 15/17 The inverse sine (sin ) is used when finding the angle while the sin is used in finding the side length. −1 *You can use a calculator to approximate the sine, cosine, and the tangent. Make sure that your calculator is in degree mode. The table shows values of each function.

Concept Development In a right triangle, the cosine of an acute angle is the ratio of the length of the adjacent leg over the length of the hypotenuse. On your whiteboard, Identify the following: What is the ratio for Cos (T)? What is the ratio for Cos (G)? In your own words, describe the sine. In Trigonometry, Cosine is_____. CFU Cos (T) = 15/17 G T R 17 8 15 Cos (G) = 8/17 The inverse sine (cos ) is used when finding the angle while the cos is used in finding the side length. −1 *You can use a calculator to approximate the sine, cosine, and the tangent. Make sure that your calculator is in degree mode. The table shows values of each function.

Hawk Dance of Miwok Indians Concept Development The Miwok Indian Tribe that at one time lived in the area which we now know as Delhi used the word Hawk Dance of Miwok Indians to help them remember the trigonometric ratios.

How are the ratios same and/or different? Skill Development/Guided Practice Trigonometric ratios are formed by comparing the lengths of two sides of a right triangle from the “viewpoint” of a given acute angle. Steps to find Trig Ratio 1 2 3 4 How did I/you know which angle to use? How did I/you identify the Opposite, Adjacent, & Hypotenuse sides? How did I/you find the unknown side length? CFU 1 2 3 Pair-share: Explain the relationship between the Sin, Cos, and Tan from #1 to #2. How are the ratios same and/or different? CFU Identify the angle to “look” through. Identify the Opp , Adj , and Hyp side to the angle. Remember the Concept Write down the Ratio. SOH-CAH-TOA Solve for the Unknown. 1. Find the sin, cos, & tan of angles A as a fraction in simplest form: 2. Find the sin, cos, & tan of angles B as a fraction in simplest form:            

Finding Sin on chart Sin 34o = 0.5592 = 0.6 Skill Development/Guided Practice Finding Sin on chart Find the sin and around to the tenths Remember the Concept Sin 34o = 0.5592 = 0.6

The tree is about 76 feet tall. Write the ratio Substitute values Relevance Reason #1: Trig Ratio are used in finding the height. You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet. tan 59° = opposite adjacent Write the ratio tan 59° = h 45 Check for Understanding Does anyone else have another reason why it is relevant to use verb tense correctly? Which reason is most relevant to you? Why? The tree is about 76 feet tall. Substitute values 45 tan 59° = h Multiply each side by 45 45 (1.6643) ≈ h Sample Item 75.9 ≈ h Simplify Find Sin, Cos, Tan of T. Leave answer as a fraction.   Relevance Reason #2: Know how to find geometric mean will help you do well on tests: (PSAT, SAT, ACT, GRE, GMAT, LSAT, etc..).    

What did you learn today about how to find the side lengths of right triangles by using the Trig Ratios. Word Bank Opposite Adjacent Arc-tangent . TOA SUMMARY CLOSURE Today, I learned how to __________________ ______________________________________________________________.