Curriculum / Math / 10th Grade / Unit 4: Right Triangles and Trigonometry / Lesson 11

Right Triangles and Trigonometry

Lesson 11

Math

Unit 4

10th Grade

Lesson 11 of 19

Objective

Find the angle measure given two sides using inverse trigonometric functions.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Similarity, Right Triangles, and Trigonometry

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Identify sine, cosine, and tangent as trigonometric functions as well as relationships between side lengths in right triangles.
Describe that because they are functions, we need to be able to UNDO them (just like addition, subtraction, etc.).
Calculate the inverse of sine, cosine, and tangent using technology—called arcsine, arccosine, and arctangent. They are also noted as $${\mathrm{sin}^{-1}(x)}$$, $${\mathrm{cos}^{-1}(x)}$$, and $${\mathrm{tan}^{-1}(x)}$$ respectively.
Use the understanding that when you give the arcsine of a value, you are saying, “What angle gives me a sine of ______?” to solve problems.
Use the inverse trigonometric functions to find the angle measures from the given ratios.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson is an extension of G-SRT.8, which does not specifically mention using inverse trigonometric function. This lesson will prepare students to access F.BF.4 standard in Algebra II.
Anchor Problem #1 will require the teacher to do a significant amount of modeling to define the inverse of sine, cosine, and tangent as arcsine, arccosine, and arctangent, respectively.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Find the value of the side length marked $$x$$.

Guiding Questions

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Problem 2

Given the following right triangle, find all of the angle measures to the nearest degree.

Guiding Questions

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Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

Include problems where students are given a real-life scenario and they must find the missing angles and side-length measures.
Include problems where students need to identify at least two ways of solving a problem—either using the Pythagorean theorem, trigonometric ratios, or inverse trigonometric ratios.
Include error analysis where the error is using the incorrect trigonometric ratio to find a missing value.

Include problems where students are given a real-life scenario and they must find the missing angles and side-length measures.
Include problems where students need to identify at least two ways of solving a problem—either using the Pythagorean theorem, trigonometric ratios, or inverse trigonometric ratios.
Include error analysis where the error is using the incorrect trigonometric ratio to find a missing value.

Mathematics Vision Project: Secondary Mathematics Two Module 6: Similarity and Right Triangle Trigonometry — Lessons 10 and 11 (Do not use any questions pertaining to angle of elevation or depression, which will be used in Lesson 12)
EngageNY Mathematics Geometry > Module 2 > Topic E > Lesson 34

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Find the measure of $${\angle CAB}$$. Round angle measures to the nearest degree. Use two different trigonometric ratios.

Lesson 10

Lesson 12

Lesson Map

Topic A: Right Triangle Properties and Side-Length Relationships

Define the parts of a right triangle and describe the properties of an altitude of a right triangle.

G.CO.A.1 G.SRT.B.4

Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.

G.SRT.B.4

Define the relationship between side lengths of special right triangles.

G.SRT.B.4 G.SRT.B.5

Multiply and divide radicals. Rationalize the denominator.

A.SSE.A.2 N.RN.A.2

Add and subtract radicals.

A.SSE.A.2 N.RN.A.2

Topic B: Right Triangle Trigonometry

Define and calculate the sine of angles in right triangles. Use similarity criteria to generalize the definition of sine to all angles of the same measure.

G.SRT.C.6

Define and calculate the cosine of angles in right triangles. Use similarity criteria to generalize the definition of cosine to all angles of the same measure.

G.SRT.C.6

Derive the relationship between sine and cosine of complementary angles in right triangles, and describe sine and cosine as angle measures approach 0°, 30°, 45°, 60°, and 90°.

G.SRT.C.7

Describe and calculate tangent in right triangles. Describe how the value of tangent changes as the angle measure approaches 0°, 45°, and 90°.

G.SRT.C.6 G.SRT.C.7

Solve for missing sides of a right triangle given the length of one side and measure of one angle.

G.SRT.C.8

Topic C: Applications of Right Triangle Trigonometry

Find the angle measure given two sides using inverse trigonometric functions.

G.SRT.C.8

Describe the relationship between slope and the tangent ratio of the angle of elevation/depression. Use the tangent ratio of the angle of elevation or depression to solve real-world problems.

G.SRT.C.8

Solve a modeling problem using trigonometry.

G.SRT.C.8

Topic D: The Unit Circle

Define angles in standard position and use them to build the first quadrant of the unit circle.

F.TF.A.2

Use the first quadrant of the unit circle to define sine, cosine, and tangent values outside the first quadrant.

F.TF.A.2

Topic E: Trigonometric Ratios in Non-Right Triangles

Derive the area formula for any triangle in terms of sine.

G.SRT.D.9

Verify algebraically and find missing measures using the Law of Sines.

G.SRT.D.10

Verify algebraically and find missing measures using the Law of Cosines.

G.SRT.D.10

Use side and angle relationships in right and non-right triangles to solve application problems.

G.SRT.D.11

Right Triangles and Trigonometry

Objective

Common Core Standards

Core Standards

Similarity, Right Triangles, and Trigonometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem Set

Target Task