Lesson 9

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.

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.)

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.

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.

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

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

Students need not complete all the problems before the discussion.

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.

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.

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.”

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. 

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.