Lesson 9

Using Trigonometric Ratios to Find Angles

9.1: Once More with the Table (10 minutes)

Warm-up

In this activity students estimate the acute angles of a right triangle with given sides and then learn how to look up the exact value in a calculator. Some students might have already estimated the angles of this triangle in a previous lesson. If so, they can check and refine their estimates now that they know how to use calculators to evaluate trigonometric ratios.

Student Facing

A triangle with side lengths 3, 4, and 5 is a right triangle by the converse of the Pythagorean Theorem. What are the measures of the acute angles?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students might struggle to start the problem. Ask them what resources they have available to help them. (Tools to draw a diagram, the right triangle table, or guess and check with the calculator.)

Activity Synthesis

Tell students that just like there is a way to ask the calculator to display the ratio for any angle in a right triangle, there is also a way to ask the calculator to display the angle for any ratio in a right triangle.

Display the image. Point out to students that we have multiple side lengths but no angle measures. “There are several possible equations we can write using \(\theta\). Let's start with \(\tan(\theta)=\frac34\). To solve this equation we need to ask the calculator what angle has a tangent of \(\frac34\).” Some calculators use arctangent and others use \(\tan^{\text-1}\) to ask this question. Demonstrate how to use the technology available in your classroom to solve \(\tan(\theta)=\frac34\) and get \(\theta=37^\circ\). Invite students to confirm that this answer matches their estimates using other methods.

A right triangle.

Tell students that the calculator produces an unreasonably long decimal value for \(\theta\). Because we have been in the habit of measuring angles using protractors which are precise to the nearest degree, we adopt the convention in these materials of rounding angle measures to the nearest whole number.

Repeat the process for angle \(C\) using a different function (arccosine or arcsine) to practice with the technology. Acknowledge that solving for the other angle using trigonometry is unnecessary due to the Triangle Angle Sum Theorem and confirm the calculations make sense by checking that the angles are complementary.

9.2: From Ratios to Angles (10 minutes)

Activity

In this activity students use trigonometry to calculate the measure of an angle when given two sides of a right triangle. Then they apply a method of their choosing to calculate other side lengths and angle measures in the triangle.

Monitor for students who use:

  • the Pythagorean Theorem
  • multiple trigonometric equations
  • the Triangle Angle Sum Theorem

Launch

“Don’t forget to calculate 3 measurements per triangle.”

Students need not complete all the problems before the discussion.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 2 of the 3 questions to complete. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Find all missing side and angle measures.

Right triangle A B C. A B is 9, C B is 10, angle B A C is 90 degrees.
Right triangle D E F. D F is 4 units and D E is 7 units.
Right triangle G H J. G H is 5 units and H J is 6 units.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Tell students there are at least 3 different methods for solving these problems:

  • Pythagorean Theorem
  • multiple trigonometric equations
  • the Triangle Angle Sum Theorem

Invite students to rank the methods in order of most to least preferred and tell them to be prepared to share their reasoning. After 1 minute of quiet think time ask a few students to share. Tell the class that there is no one best method; personal preference and the specific problem both influence your choices.

Speaking: MLR8 Discussion Supports. To support students in producing statements about the methods they preferred for solving the problems, provide sentence frames for students to use when they share. For example, “I prefer _____ because _____,” or  “_____ was better because _____.” 
Design Principle(s): Support sense-making

9.3: Leaning Ladders (15 minutes)

Activity

In this activity students will apply their knowledge of trigonometry to a real-world context. They will use the safe ladder ratio to determine the safe ladder angle. Then they can use this angle to decide if a ladder is long enough for the given scenario.

Choose if students will use the ratio \(4:1\) or measure it themselves. Note that answers will vary if students measure. The image in the task is closer to \(2:1\) than \(4:1\), so be prepared to discuss if that much variation is safe.

Launch

Tell students a ladder could tip over backwards if the angle is too large or slide down the wall if the angle is too small.

Provide students the safe ladder ratio by telling them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. On most people that ratio is approximately \(4:1\).”

Or if students will estimate the safe ladder ratio themselves, tell them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. Form groups and measure the triangle one person’s body forms. That is, measure the distance from the floor to their shoulder. Then measure the distance from the front of their chest to their wrist.”

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret the language of ratios and angles in context. Display the image and the first sentence of this problem, without revealing the questions that follow. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions with a partner before selecting 1–2 students to share their questions with the class. Listen for and amplify any questions that connect the ratios and angles to the context. 
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

A good rule of thumb for a safe angle to use when leaning a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground.

  1. What are the angles in the triangle formed by your body and the ladder?
  2. What are the angles in the triangle formed by the ladder, the ground, and the railing? Explain or show your reasoning.
  3. You have a 13 foot long ladder and need to climb to a 12 foot tall roof.
    1. If you put the top of the ladder at the top of the wall, what angle is formed between the ladder and the ground?
    2. Is it possible to adjust the ladder to a safe angle? If so, give someone instructions to do so. If not, explain why not.
Person standing with arms reached out directly in front of them touching a ladder leaning against a house.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

People have various proportions to their body. Suppose that someone’s arm to height ratio is \(5:1\).

  1. What are the angles in the triangle formed by their body and the ladder?
  2. How far off is this from the \(4:1\) safe angle?
  3. What could this person do to make the ladder closer to the safe ladder angle?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students might struggle to determine if the ladder can reach the roof. Inform them that the roof height has to be 12 feet but we can use as much of the ladder length as we want up to 13 feet. Invite them to draw a diagram where the ladder extends beyond the top of the building. 

Activity Synthesis

Display the image for all to see and invite a student to demonstrate their explanation that the triangles are similar by annotating the image. Emphasize that all of trigonometry is based on similar right triangles. These are just two of the infinite set of right triangles with a \(4:1\) ratio and an angle measure of 76 degrees.

Lesson Synthesis

Lesson Synthesis

Ask students to add these definitions to their reference charts as you add them to the class reference chart:

The arccosine of a number between 0 and 1 is the acute angle
whose cosine is that number.

$\arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right)=\theta$

A right angle, annotated.

The arcsine of a number between 0 and 1 is the acute angle
whose sine is that number.

$\arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \theta$

A right angle, annotated.

The arctangent of a positive number is the acute angle
whose tangent is that number.

$\arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right) = \theta$

A right triangle, annotated.
A right triangle.

Display triangle \(END\) for all to see. Give students 1 minute to list all the equations they can for this triangle. Invite students to share until the list includes equations using cosine, sine, and tangent as well as equations using arccosine, arcsine, and arctangent.

Ask students what information they would need to be able to calculate the rest of the values. (One angle and one side length, or two side lengths.) Tell them \(d=18, n=22\), and invite them to use any of the displayed equations to calculate the missing values. (\(e=12.6 \text{ units}, \theta=35^\circ, \alpha=55^\circ\))

9.4: Cool-down - Again with the Calculator (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Using trigonometric ratios and a calculator, the missing sides and angles of right triangles can be found.

Using the right triangle table we can estimate angle measures as in previous lessons. However, with a calculator, we can find angles more precisely.

Right triangle A B D. A C is 12 units and B C is 3 units. Angle B A C is labeled alpha and angle A B C is labeled theta.

The side opposite angle \(A\) is 3 units long, and the side adjacent to \(A\) is 12 units long. So to find angle \(A\), we write an equation using tangent: \(\tan(\alpha) = \frac{3}{12}\). To find the measure of angle \(A\) we ask the calculator, “What angle has a tangent of \(\frac{3}{12}\)?” To ask that, we use arctangent by writing \(\arctan \left(\frac{3}{12} \right)\). If we know the cosine, we use arccosine to look up the angle, and if we know the sine, we use arcsine. So \(\alpha=\arctan \left( \frac{3}{12} \right)\), which means angle \(A\) measures about 14 degrees.

Angle \(B\) can be calculated using another trigonometric equation or the Triangle Angle Sum Theorem. Let's use arctangent again. We know \(\tan(\theta)=\frac{12}{3}\), so \(\theta=\arctan \left( \frac{12}{3} \right) \), which is about 76 degrees. This matches the answer we get with the Triangle Angle Sum Theorem: \(180-90-14 = 76\).