Unit 4, Lesson 10
Using Trigonometric Ratios to Find Angles
Geometry
10.1
Warm-up
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity Narrative
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.
Launch
NoneActivity
NoneA 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
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Building on Student Thinking
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 using the calculator)
Activity Synthesis
The purpose of this discussion is to introduce arctangent, arccosine, and arcsine.
Tell students that just like there is a way to ask the calculator to display the ratio of side lengths 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 (except the 90-degree angle). “There are several possible equations we can write using . Let's start with . To solve this equation we need to ask the calculator what angle has a tangent of .” Some calculators use “arctangent” and others use “” to ask this question. Demonstrate how to use the technology available in your classroom to solve and get . 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 . Because we have been in the habit of measuring angles using protractors that are precise to the nearest degree, we will adopt the convention of rounding angle measures to the nearest whole number.
Repeat the process for angle 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.
Lesson Synthesis
Add these definitions to the class reference chart, and ask students to add them to their reference charts:
The arccosine of a number between 0 and 1 is the measure of an acute angle
whose cosine is that number. (Definition)
The arcsine of a number between 0 and 1 is the measure of an acute angle
whose sine is that number. (Definition)
The arctangent of a positive number is the measure of an acute angle
whose tangent is that number. (Definition)
Display triangle 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. (two side lengths or one angle and one side length) Tell them and , and invite them to use any of the displayed equations to calculate the unknown values. ( is 35 degrees, is 55 degrees, and is 12.6 units.)
Student Lesson Summary
Using trigonometric ratios and a calculator, the unknown side lengths and angle measures 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.
The side opposite angle is 3 units long, and the side adjacent to is 12 units long. So to find angle , we write an equation using tangent: . To find the measure of angle , we ask the calculator, “What angle has a tangent of ?” To ask that, we use arctangent by writing . The arctangent of a positive number is the measure of an acute angle whose tangent is that number. If we know the cosine of an angle, we use arccosine to look up the angle measure (the arccosine of a number between 0 and 1 is the measure of an acute angle whose cosine is that number). And if we know the sine of an angle, we use arcsine (the arcsine of a number between 0 and 1 is the measure of an acute angle whose sine is that number). So , which means angle measures about 14 degrees.
Angle can be calculated using another trigonometric equation or the Triangle Angle Sum Theorem. Let's use arctangent again. We know , so , which is about 76 degrees. This matches the answer we get with the Triangle Angle Sum Theorem: .