Curriculum / Math / 10th Grade / Unit 4: Right Triangles and Trigonometry / Lesson 15

Right Triangles and Trigonometry

Lesson 15

Math

Unit 4

10th Grade

Lesson 15 of 19

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

F.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Trigonometric Functions

F.TF.A.2 — Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Foundational Standards

The foundational standards covered in this lesson

8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:

Criteria for Success

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

Describe how the sine and cosine change depending on the quadrant you move to in the unit circle using reflections or rotations as a basis for the coordinate point changing.
Describe the transformation necessary from $${\mathrm{sin}(x)}$$ to $$-{\mathrm{sin}(x)}$$ and from $${\mathrm{cos}(x)}$$ to $$-{\mathrm{cos}(x)}$$ as reflections over the $$x$$- and $$y$$-axes, respectively.
Derive the unit circle by transforming the first quadrant reference angles across the axes.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson and the previous lesson cover key concepts in the unit by reinforcing and reviewing G-SRT.6 and G-SRT.7.
The unit circle is used throughout calculus and is built on trigonometric ratios.

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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

Given a 30° angle on the unit circle,

a) Find the coordinate point of this angle.
b) Reflect this point over the y-axis.

Guiding Questions

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Problem 2

How is $${-\mathrm{sin}(45^\circ)}$$ different from $${\mathrm{sin}(-45^\circ)}$$? Use your unit circle to help determine this.

Guiding Questions

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Problem 3

Given a 60° angle on the unit circle,

Find the coordinate point of this angle.
Find $${\mathrm{sin}(60^\circ)}$$ and $${\mathrm{cos}(60^\circ)}$$.
How could you use the value of sine and cosine to find $${\mathrm{tan}(60^\circ)}$$?

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 need to plot positive and negative angles on the unit circle. For example, “Which point on the unit circle marks -30°? What positive angle does this represent?”
Include problems where students need to recreate the unit circle given only a few values.
Include problems where students need to identify and correct an error with two trigonometric values that are congruent as well as two trigonometric values that are not congruent.
Include problems where students are given either the point, the angle, or one value and need to name the other components.
Include problems where students need to identify all of the angles that have a particular $$x$$- or $$y$$-coordinate on the unit circle. For example, “How many angles have an x-coordinate of $${\frac{1}{2}}$$? Name them and describe how you know that this is the case.”

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 4
EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 5
New Visions for Public Schools Unit 3: Trigonometric Functions — Big Idea 3, Instructional Routine: Contemplate then Calculate, "Reference Angles"
Kuta Software Free Algebra 2 Worksheets Exact Trig Values of Special Angles

Target Task

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

Below is a unit circle. The points marked on the circle represent the intersection of the terminal sides of the reference angles of 30°, 45°, and 60°. Write the coordinates for all of the marked points around the unit circle.

References

Desmos Unit Circle with Reference Triangles — Image

Lesson 14

Lesson 16

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

Trigonometric Functions

Foundational Standards

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

References