Lesson 13: Trigonometric Ratios



Unit Name: Unit 5: Similarity, Right Triangle Trigonometry, and ProofLesson Plan Number & Title: Lesson 13: Trigonometric RatiosGrade Level: High School Math IILesson Overview:Students expand their understanding of similar triangles to develop an understanding of the trigonometric ratios. Students develop an understanding of the relationship between the sine and cosine of complementary angles. Students apply their understanding of the trigonometric ratios and the Pythagorean Theorem to solve problems in which right triangles can be found. Students apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. This lesson is designed for approximately 90 to 120 minutes, but time may vary depending on the background of the students.Focus/Driving Question:How are trigonometric ratios derived from the properties of similar triangles?West Virginia College- and Career-Readiness Standards:M.2HS.46Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.M.2HS.48Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.M.2HS.49Explain and use the relationship between the sine and cosine of complementary angles.M.2HS.50Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Manage the Lesson:Through exploration and investigation, students expand their understanding of similar triangles to develop an understanding of the trigonometric ratios. Students explore the relationship between the sine and cosine of complementary angles. Students apply their understanding of the trigonometric ratios and the Pythagorean Theorem to solve problems in which right triangles can be found.Academic Vocabulary Development:cosinesinetangentIn previous lessons and courses, students understand that in similar triangles, the ratio comparing two sides of one triangle is equal to the ratio comparing the two corresponding sides in a second triangle. In the lesson launch, students examine similar right triangles and are introduced to the vocabulary that describes the ratio of two sides of a triangle: sine, cosine, and tangent. Student understanding of vocabulary related to right triangles (adjacent side, opposite side, hypotenuse) that has been previously introduced is reinforced.Launch/Introduction:Distribute 13.1 Tree Height and provide student groups to access to a variety of materials, including grid paper, protractors, compasses, rulers, etc., from which to choose as they approach the problem situation. Challenge students to find a way to determine the height of a tree. Students are given the distance that an observer is from the tree and the measure of the angle between the horizontal and his line of sight.Investigate/Explore:Distribute unlined paper to the class and ask students to use a protractor and create a large right triangle ABC with the measure of angle A = 200, the measure of angle B = 700, and the measure of angle C = 900. Ask students to carefully measure segments AB, AC, and BC to the nearest millimeter. Create a class spreadsheet for students to enter their data. After all students have entered their data, enter the formulas to determine the following ratios: opposite/hypotenuse or sine; adjacent/hypotenuse or cosine; and opposite/adjacent or tangent. Students should notice that the ratios opposite/hypotenuse or sine, adjacent/hypotenuse or cosine, and opposite/adjacent or tangent, appear equivalent (or approximately so). Students should notice that the ratios appear equivalent even though the measures of the sides of their individual triangles differ. Observe that because the triangles satisfy the AA criterion, corresponding side ratios are equal. Explain that because these ratios remain constant in similar right triangles, these ratios have been given standard names: sine, cosine, and tangent. Create a class sketch of the 20-70-90 right triangle (see diagram included in 13.2 Class Spreadsheet), identifying angle A as the 20 degree angle, angle B as the 70 degree angle, and angle C as the 90 degree angle. Ask students to first focus on angle A. Identify side AB as the hypotenuse and explain that terms need to be developed for the two legs of the triangle. Explain that the side opposite angle A is often identified with a lower case a; the side opposite angle B is often identified with a lower case b; and the side opposite angle C is often identified with a lower case c.Ask students to focus on angle A and the first ratio, BC/AB. Explain that term sine is used to identify this constant ratio. Define sin A as opposite/hypotenuse and a/c and record these ratios on 13.2 Class Spreadsheet. Ask students to again focus on angle A and the second ratio, AC/AB. Explain that term cosine is used to identify this constant ratio. Define cos A as adjacent/hypotenuse and b/c and record these ratios on the 13.2 Class Spreadsheet. Ask students to again focus on angle A and the third ratio, BC/AC. Explain that term tangent is used to identify this constant ratio. Define tan A as opposite/adjacent and a/b and record these ratios on 13.2 Class Spreadsheet. Lead students to connect the tangent ratio with the concept of slope and with the slope triangles from previous courses. (The vocabulary related to angle A and the trigonometric ratios is included in blue on the 13.2 Class Spreadsheet.)Now ask students to change their focus to angle B. On the Class Spreadsheet, create the trigonometric ratios sin B, cos B, and tan B. Ask students to identify the side opposite angle B and to create the ratio sin B (AC/AB or b/c). Include these in the Class Spreadsheet and create a formula for determining sin B. Similarly, ask students to create the ratio cos B (BC/AB or a/c) and the ratio tan B (AC/BC or b/a). Include these in the Class Spreadsheet and create a formula for determining cos B and tan B. (The vocabulary related to angle B and the trigonometric ratios is included in red on 13.2 Class Spreadsheet.)Ask students to use a calculator to find sin A (sin 20 degrees), cos A (cos 20 degrees), tan A (tan 20 degrees), sin B (sin 70 degrees), cos B (cos 70 degrees) and tan B (tan 70 degrees). Ask students to compare the class ratios with the trigonometric ratios found with the calculator. Distribute 13.3 Trigonometric Ratios of Complementary Angles. Ask students to use a calculator to find the sine, cosine, and tangent of pairs of complementary acute angles, angle A and angle B. In a class discussion, ask students what observations they can make about the trigonometric ratios. Lead students to discover that the sines and cosines of complementary angles are equal to one another. Lead students to conclude that the sin A = cos B because the side opposite angle A is the side adjacent to angle B and to similarly conclude the cos A = sin B because the side adjacent to angle A is the side opposite angle B. Explain that the “co” in cosine refers to the “sine of the complement.”Lead students to discover that tan A and tan B are reciprocals. This relationship is true by definition, since a/b and b/a are reciprocals (if a ≠ 0 and b ≠ 0). Lead students to also observe that, as the size of the acute angle increases, sines and tangents increase while cosines decrease. Introduce students to the first of two problem situations presented in 13.4 Trigonometry and Trains. In a whole class discussion lead students to apply their developing understanding of trigonometric ratios in problem situations. In the first problem situation, students investigate of the feasibility of a possible invention that prevents train collisions by allowing one train to pass over another. A second problem situation challenges students to determine the effects of a hot summer day on the expansion of railroad tracks. Lead students to understand how trigonometric ratios can be used in addressing these problem situations.In 13.5 Indirect Measurement, students will use a clinometer (a protractor-like tool used to measure angles). Several websites with directions on how to make and use a clinometer are provided in the section titled Materials. It would be advantageous for students to make and practice using their clinometers in advance of this investigation. When using a clinometer, the horizontal is not always at ground level. Indirect measurement with a clinometer often involves adding the height from the ground to the viewer’s eye level. A similar problem situation is explored in Algebra Lab, Classic Real World Examples of Trigonometric Ratios – Angle of Elevation, at . In this example, students use clinometers to determine the height of a tree. Distribute 13.5 Indirect Measurement and ask students to record their findings on the worksheet. In small groups, students use their clinometers to determine the approximate height of the flag pole (or other identified object). Encourage student groups to work together to determine the necessary measurements. Provide students with trundle wheels or measuring tapes so that they can determine their distance from the flag pole. After student groups have recorded their measurements (viewing angle, height of the observer’s eyes, and distance from observer to flag pole), in a whole class discussion, ask students if they can see a right triangle anywhere in their diagram that could be used in determining the height of the flag pole. The base of the right triangle should extend from the eyes of the observer to the flag pole. Ask how the height of the observer’s eyes affects the calculations. Lead students to realize that the two pieces of data that they have found (the length of the base of the right triangle and the measure of adjacent angle) are sufficient to create an equation involving the tangent of the angle. Solving this equation introduces the concept of inverses of trigonometric functions. Connect the use of inverse trigonometric functions to the students’ prior work with inverse functions. If necessary, provide supplementary materials to better develop student understanding of inverse functions. Create a sketch of a triangle with a 30 degree angle and an adjacent side of length 35. Ask students to create a trig function relating the information in the sketch. Students should create the equation tan 30 degrees = x/35. Ask students how they would solve this equation. Students should determine that one can solve for x by multiplying both sides of the equation by 35. Record this and ask students to use a calculator to find 35(tan 30 degrees). Students should determine that side opposite the 30 degree angle has a length of approximately 20.Ask students to work in their groups to determine the height of the flag pole. Again remind students how the height of the observer’s eyes affects the calculations.Students are also asked to use similar triangles and shadows to determine the height of the flag pole. This second method is replicates student work from a previous investigation from Lesson 9 (9.05 Me and My Shadow). Students should determine that if the measurements are taken at the same time of day that similar triangles are created by the meter stick and its shadow and the flagpole and its shadow. Therefore, the ratio comparing the height of the meter stick to the length of the meter stick’s shadow is equivalent to the ratio comparing the height of flag pole to the length of the flag pole’s shadow. Students should create and solve the proportion height of meter stick/length of meter stick’s shadow = height of flag pole/length of flag pole’s shadow. It may be beneficial to ask students to repeat the activity, finding the height of a second object, such as a football goalpost, a building, or a tall tree. Distribute 13.6 Determining Measures of a Triangle to student groups. This investigation asks students to determine the missing measures of a right triangle given the measures of an acute angle and one side, or given the measures of two sides. Students use the trigonometric ratios and the Pythagorean Theorem to find the missing measures of the right triangle. Given a right triangle where tan A = 0.75, students are asked to determine the measures of three different (similar) triangles.Distribute 13.7 Inverse Functions Examples to student groups. This investigation introduces students to the use of inverse functions in solving trigonometric equations. In the previous investigation, students were given the measure of one side and one acute angle of a right triangle and were asked to determine the measure of a second side. Students are now asked to determine the measure of an acute angle when given two sides of a right triangle. Introduce students to the idea of an inverse tangent function, sin-1. Explain that sin-1(sin y) = y and that sin(sin-1 y) = y. Ask students to use calculators to verify this inverse relationship. Ask students to verify that entering sin-1(sin 25) yields 25 and that entering sin(sin-1 42) yields 42.Summarize/Debrief:13.8 Check Your Understanding asks students to demonstrate their ability to write and interpret trigonometric ratios. Given a right triangle and the measures of the lengths of its sides, students are asked to determine the measures of its acute angles.Materials:CompassesClinometers (website directions are provided below)Grid paperMeter sticksProtractorsProtractors, drinking straws, weights, string, tape (for making clinometers)RulersTrundle wheel or Tape measures13.1 Tree Height13.1 Tree Height - Key13.2 Class Spreadsheet13.3 Trigonometric Ratios of Complementary Angles13.3 Trigonometric Ratios of Complementary Angles - Key13.4 Trigonometry and Trains13.5 Indirect Measurement13.6 Determining Measures of a Triangle13.6 Determining Measures of a Triangle - Key13.7 Inverse Functions Examples 13.8 Check Your Understanding13.8 Check Your Understanding - KeyHow to Make a Clinometer How to Make a Clinometer - Uses of a Clinometer - Algebra Lab: Classic Real World Examples of Trigonometric Ratios Triangles - Trigonometry Sine and Cosine Ratios Trigonometry Trigonometry for Solving Problems - This Illuminations lesson offers a pair of puzzles to enforce the skills of identifying equivalent trigonometric expression and problem situations involving the angle of elevation and angle of declination. Connection:As property is bought and sold, accurately determining boundary points becomes essential. Surveyors use trigonometric relationships to accurately determine the location of points and the distance and angles between them. Road makers, bridge builders and those whose job it is to get buildings in the right place all use trigonometric functions in their daily work.Lesson Reflection:Students should understand how the trigonometric ratios are derived from the properties of similar triangles. Students should understand that to solve problems involving right triangles, to determine the measures of an angle or a side of a right triangle, it is necessary to be given only the measure of two sides or one side and an acute angle.In lesson 1, teachers were provided with a guide to aid them in reflecting upon the lesson as they seek to improve their practice. Certainly, it may not be feasible to formally complete such a reflection after every lesson, but hopefully the questions can generate some ideas for contemplation. 13.1 Tree Height 13.1 Tree Height - KeyTree Height - KeyMatthew has been asked to determine the height of a tree. He measured the height from the ground to his eye and found that it was 5 feet. He used a protractor to measure the angle between the horizontal and his line of sight to the top of the tree. He found that angle to measure 70°. He also determined his distance from the tree and found that it measured 12 feet. Use the information to determine the height of the tree. Height of tree is approximately 38 feet.13.2 Class Spreadsheet 13.3 Trigonometric Ratios of Complementary AnglesTrigonometric Ratios of Complementary AnglesRecall that two angles are complementary if the sum of their measures is 90°. Use a calculator to complete the following table:Angle Asin Acos Atan AComplement of Angle A (called Angle B)sin Bcos Btan B7172430374553607180As the size of Angle A increases, what do you notice about sin A?As the size of Angle A increases, what do you notice about cos A?As the size of Angle A increases, who do you notice about tan A?What is the relationship between sin A and cos B? Why is this true?What is the relationship between cos A and sin B? Why is this true?13.3 Trigonometric Ratios of Complementary Angles KeyTrigonometric Ratios of Complementary Angles - KeyRecall that two angles are complementary if the sum of their measures is 90°. Use a calculator to complete the following table:Angle Asin Acos Atan AComplement of Angle A (called Angle B)sin Bcos Btan B70.12190.99250.1228830.99250.12198.1443170.29240.95630.3057730.95630.29243.2709240.40670.91350.4452660.91350.40672.2460300.50.86600.5774600.86600.51.7321370.60180.79860.7536530.79860.60181.3270450.70710.70711450.70710.70711530.79860.60181.3270370.60180.79860.7536600.86600.52.1445300.50.86600.5774710.94550.32562.9042190.32560.94550.3443800.98480.17365.6713100.17360.98480.1763As the size of Angle A increases, what do you notice about sin A?As the size of Angle A increases, sin A increases.As the size of Angle A increases, what do you notice about cos A?As the size of Angle A increases, cos A decreases.As the size of Angle A increases, who do you notice about tan A?As the size of Angle A increases, tan A increases.What is the relationship between sin A and cos B? Why is this true?sin A = cos B The side opposite Angle A is the side adjacent to Angle B.What is the relationship between cos A and sin B? Why is this true?cos A = sin B The side adjacent to Angle A is the side opposite Angle B.13.4 Trigonometry and Trains13.5 Indirect MeasurementIndirect MeasurementRecord your group’s measurements in the table below:Name of Students in GroupViewing AngleHeight of Observer’s EyeDistance from Observer to Flag PoleCalculated Height of Flag Pole(Method 1)Length of Shadow of Meter StickLength of Shadow of Flag PoleCalculated Height of Flag Pole(Method 2)FlagpolePersonHeight of EyesMethod 1Use your clinometers to measure the viewing angle from the horizontal to the top of the flagpole.Measure the observer’s eye height. Measure the distance from the observer to the flagpole.Place your measurements, including the viewing angle, on the diagram. Create a trigonometric ratio to calculate the approximate height of the object. How does the height of the observer’s eyes affect the calculations? Show your work.Method 2Determine the length of the shadow of the meter stickDetermine the length of the shadow of the flag pole.Create a proportion to calculate the approximate height of the object. Show your work13.6 Determining Measures of a Triangle 13.6 Determining Measures of a Triangle - Key13.7 Inverse Functions Examples13.8 Check Your Understanding13.8 Check Your Understanding - Key ................
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