# 9.3.4A Trigonometric Ratios

9-12
Subject:
Math
Strand:
Geometry & Measurement
Standard 9.3.4

Solve real-world and mathematical geometric problems using algebraic methods.

Benchmark: 9.3.4.1 Trigonometric Ratios

Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle.

Benchmark: 9.3.4.2 Solve Problems Involving Trigonometric Ratios

Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios.

For example: Find the area of a triangle, given the measure of one of its acute angles and the lengths of the two sides that form that angle.

Benchmark: 9.3.4.3 Trigonometric Ratios & Angles

Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts.

## Overview

Big Ideas and Essential Understandings
##### Standard 9.3.4 Essential Understandings

Students focusing on geometry have typically already learned and used a significant amount of algebra in their mathematics course work. This might mean they have completed at least one separate course in algebra or have studied algebra as part of an integrated approach to mathematics. Either way, they must be able to recognize the close, integral relationship between algebra and geometry and be able to use this interconnectedness to solve mathematical problems. Activities in this standard will help students do this by showing them places in their course work and also in real life where geometry and algebra are used together in order to solve problems.

In this standard, students will connect algebra and geometry in a number of ways. They will calculate distances and angle measures using trigonometric ratios. They will look at slope of a line, midpoint of a segment and length of a segment using formulas derived from coordinate geometry. They will calculate geometric transformations through the use of the Cartesian plane. In addition, they will graph diameter and circumference of round objects, in order to see a visual interpretation of the number pi.

##### All Standard Benchmarks

9.3.4.1
Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle.
9.3.4.2
Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios.
9.3.4.3
Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts.
9.3.4.4
Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments.
9.3.4.5
Know the equation for the graph of a circle with radius r and center (h, k), (x - h)2 + (y - k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations.
9.3.4.6 Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90˚, to solve problems involving figures on a coordinate grid.
9.3.4.7
Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure.

Benchmark Cluster

#### Benchmark Group A - Trigonometric Ratios

##### 9.3.4

9.3.4.1
Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle.
9.3.4.2
Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios.
9.3.4.3
Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts.

##### What students should know and be able to do [at a mastery level] related to these benchmarks:
• Determine which sides of a right triangle are used to form each trigonometric ratio for a given acute angle;
• Write the ratio for the sine, cosine and tangent of an acute angle in a right triangle;
• Use the ratio for the sine, cosine, or tangent of an acute angle in a right triangle to determine side lengths of the triangle;
• Evaluate sine, cosine and tangent using a calculator;
• Use the measurement of an acute angle and one side length to calculate the other two side lengths in a right triangle;
• Given two sides of a right triangle, use a trigonometric inverse to determine angle measures.
##### Work from previous grades that supports this new learning includes:
• Use linear equations to represent proportional relationships.
• Use the Pythagorean Theorem to solve problems involving right triangles.
• Solve for missing sides and angles of similar triangles.
Correlations

#### NCTM Standards

##### Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:

• Analyze properties and determine attributes of two- and three-dimensional objects;
• Use trigonometric relationships to determine lengths and angle measures.

Specify locations and describe spatial relationships using coordinate geometry and other representational systems:

• Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations;
• Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.

Apply transformations and use symmetry to analyze mathematical situations:

• Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices;
• Use various representations to help understand the effects of simple transformations and their compositions.
##### Common Core State Standards (CCSS)

HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Define trigonometric ratios and solve problems involving right triangles.

• HS.G-SRT.8. Understand 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.

HS.G-SRT (Similarity, Right Triangles, & Trigonometry) Apply trigonometry to general triangles.

• HS.G-SRT.9. Explain and use the relationship between the sine and cosine of complementary angles.
• HS.G-SRT.10. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

HS.G-GPE (Expressing Geometric Properties with Equations) Translate between the geometric description and the equation for a conic section.

• HS.G-GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

HS.G-GPE (Expressing Geometric Properties with Equations) Use coordinates to prove simple geometric theorems algebraically.

• HS.G-GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
• HS.G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

HS.G-CO (Congruence) Experiment with transformations in the plane.

• HS.G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
• HS.G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
• HS.G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
• HS.G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

HS.A-CED (Creating Equations) Create equations that describe numbers or relationships.

• HS.A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

## Misconceptions

Student Misconceptions
##### Student Misconceptions and Common Errors
• Students get confused about which side of the triangle is the "opposite" and which is the "adjacent," especially when switching between the acute angles.
• Students confuse which trigonometric ratio works for their written proportion.
• Students rely on using SOH-CAH-TOA to remember the trigonometric ratios and then misspell the mnemonic and become more confused.
• Students attempt to use the right angle with the trigonometric definitions and identify the hypotenuse as the "opposite."
• Students do not understand that their calculators can evaluate trigonometric ratios for different angle measurement systems and forget to check that they are using the correct mode.
• Student's misuse cross-multiplication when solving trigonometric equations and invert portions of their answers.

## Vignette

##### In the Classroom

This is an activity that helps students make estimates of distance - specifically, the heights of buildings and structures on and/or near the school or its grounds. In addition, they use trigonometry to compute the height of one structure on the school grounds.

Each student pair will need a notebook, pencil, estimation worksheet (see below), yardstick or meter stick and large protractor (or clinometer). In addition, the teacher will need a 100-foot tape measure.

If students don't have access to a large, "chalkboard-style" protractor, they can make their own clinometer.

Teacher: Today, we're going to go on a field trip. We're going to go outside and estimate heights of a few objects near the school. When we're done estimating, we'll take some measurements of angles and relate those measurements to how we can figure out if our estimates are close.

Be sure to bring your estimation worksheets, as well as your notebooks, pencils and protractors.

This worksheet is based on measuring a structure that has two different "landmarks" - one is the rotor of a wind turbine, and the other is the top of the turbine. For this activity, students estimated the heights of both. Your students might estimate the height of any number of structures, depending on the landmarks at or near your school.)

##### Estimation Worksheet
 Height of school building Height of school building with person as "hint" Height to rotor (first location) Total height of turbine (first location) Height to rotor (second location) Total height of turbine (second location) Height to rotor (third location) Total height of turbine (third location) Actual height to rotor Actual total height of turbine Angle of elevation to rotor using protractor Angle of elevation to top of turbine using protractor Actual angle of elevation to rotor Actual angle of elevation to top of turbine

Begin by taking students on a "field trip" outside, to estimate the height of their school building.

Students should be approximately 50-100 feet from the building when estimating. They might wish to have another student (or adult) stand next to the building when they make their estimates, in order to give some perspective to the height.

It would help if the teacher knew in advance the height of the building at the particular spot(s) they go to, but if not, perhaps a custodian could go up on the roof with a tape measure to take a measurement.

Next, the students should focus on the location of the first structure (tree, goal posts, building, wind turbine, etc.) they wish to estimate the height of, and head in that general direction. Eventually, the students will work with their protractors and yardsticks when they're 100 feet away from the structure, but they should make one or two estimates along the way.

For example, they might be about 400 meters from a structure, and make an estimate there. Then, they will move to about 200 meters from the structure, and make another estimate of the height. The purpose of this is to see if the students' estimates change as they get closer to the object.

Finally, the students should move to a spot 100 feet from the structure and make another, final estimate, possibly with a person standing in front of it, in order to give some idea of scale.

At this point, once the students have made their second (or third) estimate of the height of the structure, the teacher will use the 100-foot tape measure to mark off a distance of 100 feet from the structure. The students will then use their protractors and yardsticks to measure the approximate angle of elevation to the top of the structure.

The students will work in pairs, with one student holding the yardstick (in a way similar to the picture below) so that the other can measure the angle to the top.

After the students record their estimates, the students and teacher will go back to the classroom and perform the necessary calculations to find the height of the structure, according to the angle estimates they made.

Teacher: What was your estimate of the angle to the rotor?

Student 1: We got 35 degrees.

Student 2: We got 46 degrees.

Student 3: We got 33 degrees.

Student 4: We got 138 degrees.

Student 2: How could it be that different? All of our measurements were acute angles, and that's an obtuse angle.

Student 4: Maybe we read the protractor incorrectly.

Teacher: If that's the case, what was your "real" estimate?

Student 4: It would be 180 - 138 degrees, or 42 degrees. That sounds more reasonable.

Teacher: Will these differences in estimates matter, given that you're 100 feet from the turbine?

Student 1: Yes, they'll make a big difference in what we get for the height.

Teacher: Is there a way we could make the measurements standard?

Student 2: How about if we take the average of all the measurements?

Student 3: That's a good idea.

The students calculate an angle of elevation of 38 degrees to the rotor.

Teacher: How can we use this measure to calculate the height to the rotor?

Student 4: Could we use trigonometry?

Teacher: In what way would you use it?

Student 4: I'm not sure. How can we tell which function to use?

Student 1: I think I know. If the angle of elevation is the angle we're using, then the height of the rotor would be the opposite leg, and the 100 feet that we are from the rotor would be the adjacent leg, so we would use the tangent.

Teacher: OK, now what would your equation look like?

Student 2: We know that the tangent ratio is the length of the opposite leg divided by the length of the adjacent leg, so it would look like this:
tan 38o = $\frac{h}{100}$

If you multiply both sides by 100, then 100tan 38 = h, and the height is approximately 78.1 feet.

Teacher: Does that seem reasonable to you?

Student 3: Wait! We held the protractors at about our eye level. Would that make any difference?

Teacher: In what way would that matter?

Student 3: Well, wouldn't we have to add about five feet to the measure, because our eye levels are at about five feet?

Teacher: Good thinking. Is that reasonable to you?

Student 3: I think so. That would make our height about 83.1 feet.

Teacher: Yes, and the actual height is 80 feet. How do you think your estimate compared to the actual height?

Student 5: I think it was pretty close.

Teacher: Me, too. Good job. Let's look at one other aspect of this. We know that the height to the rotor is 80 feet, and we know that we estimated the angle from a distance of 100 feet from the structure. How could we calculate the actual measure of the angle of elevation from the ground to the rotor?

Student 6: Would we have to use an inverse function?

Teacher: Why do you think you'd have to do that?

Student 6: Because we use those when we're looking for angle measures.

Teacher: Excellent. Which inverse function would you use?

Student 6: I think it's the inverse tangent function, because we're working with the opposite leg and the adjacent leg.

Teacher: Right. What would your equation look like?

Student 7: I think I know. If we're looking to find the measure of angle A, then I think it would be:

tan-1 A = $\frac{80}{100}$

Then the measure of angle A would be about 38.7 degrees.

## Resources

Instructional Notes
##### Teacher Notes
• Teachers should model identifying which acute angle is to be used first before considering sides of the triangle.
• Teachers need to be clear about which trigonometric relationship applies to the situation and explain why. If the mnemonic SOH-CAH-TOA is to be used, model using it every time.
• Teachers need to clarify that these trigonometric definitions only apply to acute angles in right triangles.
• Teachers may introduce the concept of radian measurement so that students know there are other units possible for measuring angle magnitude.
• Teachers may model cross-products for solving proportions and avoid using the cross-multiplication short-cut.
Instructional Resources

Law of Sines and the Law of Cosines
In this unit, students learn the Law of Sines and the Law of Cosines and determine when each can be used to find a side length or angle of a triangle.

National Council of Teachers of Mathematics (NCTM). (2010). Focus in High School Mathematics: Reasoning and Sense Making in Geometry. Reston, VA: National Council of Teachers of Mathematics.

Basic trigonometry

New Vocabulary

trigonometry: the study of triangles and the relationships between their sides and angles.

sine: the sine of an acute angle in a right triangle is the ratio of the length of the opposite leg to the length of the hypotenuse.

cosine: the cosine of an acute angle in a right triangle is the ratio of the length of the adjacent leg to the length of the hypotenuse.

tangent: the tangent of an acute angle in a right triangle is the ratio of the length of the opposite leg to the length of the adjacent leg.

Professional Learning Communities
##### Reflection - Critical Questions regarding the teaching and learning of these benchmarks
• What other instructional strategies can be used to engage students' study of angles associated with parallel and perpendicular lines?
• How can manipulatives be used to help students visualize these abstract concepts?
• How can the instruction be scaffolded for students?
• What additional scaffolding is needed to provide ELL students?
• Do the tasks that have been designed connect to underlying concepts or focus on memorization?
• How can it be determined if students have reached this learning goal?
• How can the lesson be differentiated?

## Assessment

This Assessment references the Vignette Activity for this Framework.

1. The minimum angle measure to a wind turbine's rotor was 32 degrees, and the maximum angle measure was 47 degrees. What would be the minimum and maximum heights to the rotor with these angle measurements?

2. If the average angle of elevation to the top of the turbine was calculated to be 49.5 degrees, according to the calculations, what is the total height of the wind turbine?

3. If the range of angle measures was 45 to 58 degrees, what would be the minimum and maximum heights to the top of the turbine?

4. If the height to the top of the structure is 115 feet, what is the angle of elevation from the ground when measured at a distance of 100 feet from the structure? Call this angle B.

##### Assessment solutions

1. DOK Level: 3

Since the problem is working with the opposite leg (the height to the rotor) and the adjacent leg (the distance from the point of measurement to the turbine), the tangent function should be used. If the angle measure is called m, and the height to the rotor is called h, then

tan mo = $\frac{h}{100}$

Plugging in 32 degrees and 47 degrees into the above equation, the result is a minimum height of 67.5 feet to the rotor (after adding five feet), and a maximum height of 112.2 feet to the rotor (after adding five feet for the distance the protractor was held off the ground).

2. DOK Level: 2

By plugging 49.5 degrees into the equation, the result is a total height of 122.1 feet (after adding five feet). The actual height to the top of the turbine is 115 feet.

3. DOK Level: 3

Plugging these angle measures into the equation (and adding five feet), the result is a minimum height of 105 feet and a maximum height of 165 feet.

4. DOK Level: 2

The result is:

tan-1 B = $\frac{115}{100}$

So the measure of angle B would be approximately 49 degrees.

## Differentiation

Struggling Learners
• Split the Vignette activity into two or more separate tasks. One way to do this would be to have students make their height estimates first, then do the estimates of the angles and finally do the computations.
English Language Learners
• Vocabulary is an issue for ELL students. Spend time in class before the Vignette activity reviewing relevant terminology and showing pictures as needed.
• Review measurements in feet (as opposed to meters or other metric units), and review terminology relevant to a protractor and/or clinometer.
Extending the Learning
• Extend the Vignette activity by having pairs of students make clinometers and compare their results with the results when using the protractors.