Introduction to Trigonometry

Transcription

1 NAME COMMON CORE GEOMETRY- Unit 6 Introduction to Trigonometry DATE PAGE TOPIC HOMEWORK 1/ Lesson 1 : Incredibly Useful Ratios Homework Worksheet 1/ LESSON 2: Using Trigonometry to find missing sides Homework Worksheet 1/ LESSON 3: Using Trig to find missing angles Homework Worksheet measures 1/25 Review/ Quiz No homework 1/ LESSON 4: Angles of Elevation and Depression Homework Worksheet 1/ LESSON 5: Special Right Triangles Homework Worksheet 1/30 Class Work Finish Class Work 1/ LESSON 6: Practice with Regents Questions Review Worksheet 2/1 Review Ticket In 2/2 Test 1

2 Lesson 1: Incredibly Useful Ratios For each triangle, label the appropriate sides as hypotenuse, opposite, and adjacent with respect to the marked acute angle Vocabulary o The side of a right triangle opposite the right angle is called the. o The leg of a right triangle across from the marked acute angle is called the side. o The leg of a right triangle (one of the two rays of the marked acute angle) is called the side. In trigonometry, we sometimes represent the measure of the angle with the Greek letter θ, pronounced theta. If is the angle measure of A as shown, then we define: The sine of is: As a formula, sin In the given diagram, sin A The cosine of is: As a formula, cos In the given diagram, cos A The tangent of is: As a formula, tan In the given diagram, tan A 2

3 Examples Label the sides of each triangle with respect to the circles angles as: Hyp/Adj/Opp 1. Using PQR, complete the following table. (Do not simplify the ratios.) name sine opp hyp cosine adj hyp tangent opp adj P Q 2. Ashlyn did not finish completing the table below for a diagram similar to the previous problems that we just completed. In the diagram, p represents the measure of P and q represents the measure of Q. Use any patterns you notice from Exercises 1 to complete the table for Ashlyn. Measure of Angle Sine Cosine Tangent p q sin 11 6 p cos p tan p Consider the right triangle ABC where C is a right angle. a. Find the sum of A B. b. Find the ratios for sin A and cos B. What do you notice? c. Find the ratios for cos A and sinb. What do you notice? 3

4 Important Discovery! The acute angles in a right triangle are always complementary. The sine of any acute angle is equal to the cosine of its complement. A B 90 iff sina cosb Using the 2 equations listed above, how can we rewrite sina=cosb in terms of ONE angle? 2. Find the values for that make each statement true. sin cos 25 b. sin80 cos a. c. sin cos 10 d. sin 45 cos 3.) In right triangle ABC with right angle at C, sina=2x+0.1 and cosb=4x-0.7. Determine and state the value of x. 4

5 Lesson 2: Using Trigonometry to Find Side Lengths Recall the three trig ratios: sin cos tan 2. Consider the given triangle. a. Using trig ratios, find the length of side a to the nearest hundredth. b. Now calculate the length of side b to the nearest hundredth. c. Could we have used another method to determine the length of side b? If so, what is it? 3. Given right triangle GHI, with right angle at H, GH 12.2 and G 28. Find the measures of the remaining sides and missing angle measure rounded to the nearest tenth. 5

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7 Lesson 3: Using Trigonometry to Find Angle Measures Using Trigonometry to Find Angle Measures: -In Algebra, we could solve 2x 14 by doing the opposite of multiplying by 2, which is dividing. -We also solved 2 x 9 by doing the opposite of squaring, which is taking the square root. -In trigonometry, to solve 1 sin x 2 we need to do the opposite of sin, which is arcsin. CALCULATOR TIPS: To solve in your calculator: o o check that your mode is in DEGREES turn the equation into Be sure to show this work on your paper! o which is the same thing as o press and your calculator will display o type in as using the division key o hit enter to see the angle measure that has a sine value of You will need a calculator Example 1 Find the angle measure from the boy to the top of the tree. Round your answer to the nearest hundredth

8 Example 2 Find the measure of a to the nearest degree. Example 3 Find the measure of b to the nearest degree. Example 4 A 16 foot ladder leans against a wall. The foot of the ladder is 7 feet from the wall. a. Find the vertical distance from the ground to the point where the top of the ladder touches the wall. Round your answer to the nearest tenth. b. Determine the measure of the angle formed by the ladder and the ground. Round your answer to the nearest degree. Exercises 1. Find the measure of c to the nearest degree. 2. Find the measure of d to the nearest degree. 8

9 3. A roller coaster travels 80 ft of track from the loading zone before reaching its peak. The horizontal distance between the loading zone and the base of the peak is 50 ft. At what angle, to the nearest degree, is the roller coaster rising? 9

10 Lesson 4: Angles of Elevation and Depression Example 1 The angle of elevation from a point 25 feet from the base of a tree on level ground to the top of the tree is 30. Draw a picture to model the situation, and then find the height of the tree to the nearest tenth of a foot. Example 2 Standing on the gallery of a lighthouse (the deck at the top of a lighthouse), a person spots a ship at an angle of depression of 20. The lighthouse is 28 m tall and sits on a cliff 45 m tall as measured from sea level. What is the horizontal distance between the lighthouse and the ship? Round your answer to the nearest whole meter. 10

11 Example 3 Samuel is at the top of a tower and will ride down a zip line to a lower tower. The total vertical drop of the zip line is 40 ft. The zip line s angle of elevation from the lower tower is To the nearest tenth, what is the horizontal distance between the towers? Example 4 An anchor cable supports a vertical utility pole forming a 51 angle with the ground. The cable is attached to the top of the pole. If the distance from the base of the pole to the base of the cable is 5 meters, how tall is the pole rounded to the nearest hundredth? Example 5 11

12 Lesson 5: Special Right Triangles Opening Exercise There are certain special angles where it is possible to give the exact value of sine and cosine. These frequently seen angles are 0,, 45,, and Using the given triangles, complete the following table and rationalize the denominators if necessary. θ Sine 0 1 Cosine 1 0 Study this chart and make some observations about the ratios, develop a strategy to memorize these important trig angles and write it below: Ratio of Sides of Special Right Triangles triangle triangle 2 : 2 3 : 4 2 : 2 : : : 3 : : 4 : : 4 : : x : : x : : 12

13 Exercises 1. Find the exact value of the missing side lengths in the given triangle. Start by drawing the similar right triangle from the previous page: 2. Find the exact value of the missing side lengths in the given triangle. 3. Find the exact value of the missing side lengths in the given triangle. a b 6 13

14 LESSON 6- REGENTS TRIG QUESTIONS 1.) In, where is a right angle,. What is? 1) 2) 3) 4) 2.) Which expression is always equivalent to when? 1) 2) 3) 4) 3.) In the diagram below, a window of a house is 15 feet above the ground. A ladder is placed against the house with its base at an angle of 75 with the ground. Determine and state the length of the ladder to the nearest tenth of a foot. 4.) As modeled below, a movie is projected onto a large outdoor screen. The bottom of the 60-foot-tall screen is 12 feet off the ground. The projector sits on the ground at a horizontal distance of 75 feet from the screen. Determine and state, to the nearest tenth of a degree, the measure of, the projection angle. 14

15 5.) As shown in the diagram below, a ship is heading directly toward a lighthouse whose beacon is 125 feet above sea level. At the first sighting, point A, the angle of elevation from the ship to the light was 7. A short time later, at point D, the angle of elevation was 16. To the nearest foot, determine and state how far the ship traveled from point A to point D. 6.) Cathy wants to determine the height of the flagpole shown in the diagram below. She uses a survey instrument to measure the angle of elevation to the top of the flagpole, and determines it to be She walks 8 meters closer and determines the new measure of the angle of elevation to be At each measurement, the survey instrument is 1.7 meters above the ground. Determine and state, to the nearest tenth of a meter, the height of the flagpole. 15