Illustrative Mathematics Geometry, Unit 4.6 - Teachers (2024)

Estimate the value of \(z\).

Tell students, “The right triangle table is useful, but what if the angle is not a multipleof 10 degrees? There must be a way to move from estimating to calculating.The name of the column in the right triangle table, 'adjacent leg divided by the hypotenuse,'is long, so mathematicians call this ratio cosine. The column called 'opposite leg divided by the hypotenuse'is named sine. The column called 'opposite leg divided by the adjacent leg'is named tangent. These three names describe trigonometric ratios.Label the columns of the right triangle table with the corresponding trigonometric function names.”

“The word trigonometric comes from the ancient roots of trigon (think about what pentagon or hexagon mean) and metric (like measure). So the word trigonometriccomes from ancient words meaning triangle measurement.”

“Cosine, sine, and tangentwork like functions where the input is the measure of an angle and the output is a ratio. Scientific calculators can display the ratio for any angle, even ones not included in the table. First, identify that we have an angle measure and the length of the adjacent leg. Then identify that we are looking for the length of the hypotenuse. The ratio of adjacent leg divided by hypotenuse is called cosine. So we can write \(\cos(50)=\frac{3}{z}\).” Demonstrate how to use the technology availableto solve this type of problem. Typing \(\cos(50)\) into the calculator and then substituting gives the new equation \(0.643=\frac{3}{z}\). (Note: if students don’t get 0.643, check the mode of the calculator. There is no need to discuss the meaning of radians now, but students will see radians later in this course.)

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a scientific calculator. Some students may benefit from a checklist or list of steps to be able to use the calculator to solve trigonometric functions.
Supports accessibility for: Organization; Conceptual processing; Attention

Invite students to share their new value for \(z\). (Solving theequation we wrote in the launch gives \(z=4.7\) units.)

Ask students:

• “How does this answer compare to using the table?”(It’s the same.)
• “Which one is more trustworthy?” (All opinions are valid.)
• “Which one is more accurate?” (The calculator is much more accurate than the table for numbers between decade numbers. The calculator gives more decimal places, and it’s possible to solve without rounding to get a more precise answer.)

Inform students it is also okay to leave answers in the form \(z=\frac{3}{\cos(50)}\)which would be exact.

Ask students to add thesedefinitionsto their reference charts as you add themto the class reference chart:

The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse.

\(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\)

The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse.

\(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\)

The tangentof an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg.

\(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\)

Speaking: MLR8 Discussion Supports. Use this to routine to provide students with multiple opportunities to produce the language of cosine, sine, and tangent. As students add to their reference charts, read each definition aloud. Invite all students to repeat each definition in unison 1–2 times. Choral repetition facilitates the transfer of new vocabulary to students’ long term memory.
Design Principle(s): Maximize meta-awareness

Students apply their new knowledge of trigonometric ratios to solve these problems. Since these problems ask for multiple side and angle measures, there is an increased opportunity for creativity in solving.

Monitor for students who solve multiple trigonometric equations versus those that apply the Pythagorean Theorem.

Arrange students in groups of 2. Ask students to compare their strategywith their partner’s and decide if they are both correct, even if they are different.

Students need not complete all the problems before the discussion.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students 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

1. Solve for \(x\).

   

2. Solve for \(y\).

   

3. Find all the missing sides and angle measures.

   1. The measure of angle \(X\) is 90 degrees and angle \(Y\) is 12 degrees. Side \(XZ\) has length 2 cm.

   2. 

   3. The measure of angle \(K\) is 90 degrees and angle \(L\) is 71 degrees. Side \(LM\) has length 20 cm.

Complete the table.

Based on this information, what do you think are the cosine, sine, and tangent of 90 degrees? Explain or show your reasoning.

Invite students to contrast solving multiple trigonometric equations versus applying the Pythagorean Theorem to determine the final side length. Is one method easier? More accurate? (The Pythagorean Theorem is more familiar so I prefer it. Both methods are accurate so long as you don't round too much.)

Call attention to the triangle with no marked right angle. Ask students, “How do you know you can use trigonometric functions with this triangle?” (By the Triangle Angle Sum Theorem the third angle must be 90 degrees.)

Display this triangle and give students quiet work time to write as many equations as they can to show the relationships in triangle \(END\).

Invite students to share one equation at a time until you have recorded two of each trigonometric ratio (cosine, sine, and tangent):

Tell students the value of \(d\) is 6 units. First, ask them to consider which equations would be helpful in solving for the other sides. Then give them time to do the calculations(\(e=4.2 \) units, \(n=7.3\)units). If students get slightly different answers, take the opportunity to discuss precision and rounding error.

We have a column in the right triangle table for "adjacent leg\(\div\)hypotenuse."We use this ratio so frequently it has a name. It is called the cosine of the angle. We write \(\cos(25)\) to say the cosine of 25 degrees. A scientific calculator can display the cosine of any angle. This means we can more precisely calculate unknown side lengths rather than estimating using the table. The right triangle table is sometimes called a trigonometry table since cosine, sine, and tangent are trigonometric ratios. Here is what the table looks like with the ratios labeled with their special names:

If the length\(b\)is 7, we can find \(c\) by solving the equation \(\cos(25)=\frac{7}{c}\). So \(c\)is about 7.7 units. To solve for \(a\) we have 3choices: the Pythagorean Theorem, sine, and tangent. Let’s use tangent by solving the equation \(\tan(25)=\frac{a}{7}\). So \(a\)is about 3.3units. We can check our answers using the Pythagorean Theorem. It should be true that \(3.3^2+7^2=7.7^2\). The two expressions are almost equal, which makes sense becausewe expect some error due to rounding.