Trigonometric Ratios and Special Right Triangles

1. ACT RESPONSIBLY & SUPPORT the COMMUNITY. • Be on Time • Wear ID • Chromebook Ready • SEE SOMETHING, SAY SOMETHING

2. ° Learning Objective We will determine1 how to use the Special Right Triangle to determine and find the side lengths. What are we going to do? What is determine 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(Hyp )- the longest side of a right triangle, opposite the right angle (90 ). Hypotenuse Opposite o Adjacent(Adj) -next to something else. Adjacent Use a Trigonometric Ratio to find the value of x. Students, you already know identify the sides in a right triangle. Today, we will learn how to use the Special Right Triangle to determine and find the side lengths.. Make Connection Hypotenuse 1 Figure out Vocabulary Opposite x = 37

3. Concept Development A trigonometric ratio is a ratio of two sides of a right triangle. You have already seen SOH-CAH-TOA, which represents all three ratios. Recall: The point (x, y). The x will now be represented by Cos and ywill be represented by Sin. CFU • On your whiteboard, Identify the following using the Unit Circle: • cos 30° = • 2. sin 30° = • 3. cos 45° = • 4. sin 45° = • 5. cos 60° = • 6. sin 60° = (0, 1) (1, 0) (-1, 0) (0, -1)

4. Concept Development In geometry, an isosceles triangle is a triangle that has two sides of equal length.  Pythagorean Theorem CFU • On your whiteboard, Find the indicated values from the figure. • RT = • RS = • Sin 30° = • Tan 60° = Find the given side lengths and angle measurements for triangle ABC.

5. 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.

6. 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. 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 SOH-CAH-TOA Identify the angle to “look” through. Steps to find Trig Ratio 1 Identify the Opp, Adj , and Hyp side to the angle. 1 2 Write down the Ratio. 2 Solve for the Unknown. 3 3 4 1. For the triangle, find the unknown side lengths and trigonometric ratios for the 45° angles. 2. Find the unknown side lengths in each right triangle.

7. opposite tan 59° = adjacent h tan 59° = 45 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. Write the ratio • 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 Trig Ratios will help you do well on tests: (PSAT, SAT, ACT, GRE, GMAT, LSAT, etc..).

8. What did you learn today about how to use the Special Right Triangle to determine and find the side lengths. Word Bank SUMMARY CLOSURE • Unit Circle • 45-45-90 Triangle • 30-60-90 Triangle • . Today, I learned how to __________________ ______________________________________________________________.