Lesson 6

Working with Trigonometric Ratios

6.1: This Time with Strategies (5 minutes)

Warm-up

This warm-up uses a problem students didn’t know how to approach at the beginning of the unit. At this point they can try again using the right triangle table. Then when students learn to look up trigonometric ratios in the calculator, that procedure will be solidly grounded in conceptual understanding.

Student Facing

Estimate the value of \(z\).

Right triangle GHJ. Angle G = 90 degrees, angle J= 40 degrees, angle H= 50 degrees. Side GH=3 units. Side HJ labeled z.

Student Response

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

Activity Synthesis

Point out to students this is the triangle they didn’t know how to approach at the beginning of the unit. Now they have a strategy for finding side lengths of right triangles for acute angles included in the right triangle table. The purpose of this lesson is to teach a method that works for any angle, not just the ones in the table.

6.2: New Names, Same Ratios (15 minutes)

Activity

The purpose of this activity is to connect the work students have done in previous lessons and the warm-up to trigonometric ratios. Students learn the names of the trigonometric ratios and how to look them up in the calculator. To continue solidifying their conceptual understanding, students compare the calculator’s value to their work with the right triangle table.

Launch

Tell students, “The right triangle table is useful, but what if the angle is not a multiple of 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 trigonometric comes from ancient words meaning triangle measurement.”

“Cosine, sine, and tangent work 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 available to 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

Student Facing

  1. Use your calculator to determine the values of \(\cos(50)\)\(\sin(50)\), and \(\tan(50)\).
  2. Use your calculator to determine the values of \(\cos(40)\)\(\sin(40)\), and \(\tan(40)\).
  3. How do these values compare to your chart? 
  4. Find the value of \(z\).
Right triangle GHJ. Angle G = 90 degrees, angle J= 40 degrees, angle H= 50 degrees. Side GH=3 units. Side HJ labeled z.

Student Response

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

Activity Synthesis

Invite students to share their new value for \(z\). (Solving the equation 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 these definitions to their reference charts as you add them to 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.

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

A right triangle, annotated.

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.

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

A right triangle, annotated.

The tangent of 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.

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

A right triangle, annotated.
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

6.3: Solve These Triangles (15 minutes)

Activity

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.

Launch

Arrange students in groups of 2. Ask students to compare their strategy with 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

Student Facing

  1. Solve for \(x\).

    Right triangle ABC. Angle A= 90 degrees, angle B=17 degrees. Side AB = 10 units. Side BC labeled x.
  2. Solve for \(y\).

    triangle DEF. Angle D = 42 degrees, angle E = 48 degrees. Side DE=7 units. Side DF labeled 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. Right triangle N P Q. N Q is 4 units, angle N Q P is 90 degrees, angle P N Q is 60 degrees.
    3. The measure of angle \(K\) is 90 degrees and angle \(L\) is 71 degrees. Side \(LM\) has length 20 cm.

Student Response

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

Student Facing

Are you ready for more?

Complete the table.

angle cosine sine tangent
\(80^\circ\)      
\(85^\circ\)      
\(89^\circ\)      

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

Student Response

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

Anticipated Misconceptions

If students struggle to get started, prompt them to set up ratios to find the missing sides. If students struggle to set up ratios, prompt them to identify what is known and what they are looking to find by annotating the diagram.

Activity Synthesis

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.)

Lesson Synthesis

Lesson Synthesis

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

A right triangle.

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

\(\cos(35)=\frac{d}{n}\)

\(\sin(35)=\frac{e}{n}\)

\(\tan(35)=\frac{e}{d}\)

 

\(\cos(55)=\frac{e}{n}\)

\(\sin(55)=\frac{d}{n}\)

\(\tan(55)=\frac{d}{e}\)

 

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.

6.4: Cool-down - Solve That Triangle (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

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:

  cosine sine tangent
angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
\(25^\circ\) \(\cos(25)=0.906\) \(\sin(25)=0.423\) \(\tan(25)=0.466\)
Right triangle. Sides labeled a,b,c. Angle opposite side c= 90 degrees, angle opposite side a = 25 degrees.

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 3 choices: the Pythagorean Theorem, sine, and tangent. Let’s use tangent by solving the equation \(\tan(25)=\frac{a}{7}\). So \(a\) is about 3.3 units. 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 because we expect some error due to rounding.