Lesson 14 | Right Triangles and Trigonometry | 10th Grade Mathematics

Objective

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

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

4.G.A.1

Geometry

4.G.A.1 — Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Criteria for Success

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

Identify an angle in standard position if the vertex is at the origin and one ray is on the positive $$x$$-axis.
Orient the initial side with the ray on the $$x$$-axis and the terminal side as the other ray.
Describe the unit circle as a circle where the radii are one unit and each radius is the hypotenuse of triangles formed inside the unit circle.
Describe the importance of the unit circle as it relates to trigonometric ratios. This unit circle will be used throughout higher math.
Define reference angles as positive and only in the first quadrant.
Know that $$(x_\theta,y_\theta)=(\mathrm{cos}\theta,\mathrm{sin}\theta)$$

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson and the next 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 blank coordinate plane, draw a 45° angle with a vertex at the origin.

Guiding Questions

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Problem 2

Suppose that point $$P$$ is the point on the unit circle obtained by rotating the initial ray 30°. Find $${\mathrm{sin}(30^\circ)}$$ and $${\mathrm{cos}(30^\circ)}$$.

Guiding Questions

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References

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 4 — Example 1

Algebra II > Module 2 > Topic A > Lesson 4 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 3

Each of the points in Quadrant I represent THE point of intersection of a ray that forms a 30° angle, a 45° angle, and a 60° angle with the $$x$$-axis in the initial position. What are the coordinate points where the rays intersect the circle at 30°, 45°, and 60°?

Guiding Questions

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References

Desmos Unit Circle with Reference Triangles

Target Task

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

Draw an angle in standard position that has an intersection point on the unit circle of $${\left(\frac{1}{2},\frac{\sqrt3}{2}\right)}$$. What is the sine of this angle? The cosine?

Additional Practice

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

Include problems where students are asked to identify and fix the error. For example, give the sine of an angle and a drawing on the unit circle with the angle not in standard position.
Include problems that require students to find the coordinate point based on the angle provided. For example, given an angle, find the coordinate point. Make sure to vary the information given and what is requested for an answer.
Include problems where students need to make a drawing based on the parameters provided. For example, “Draw an angle that has a sine of… Is there only one angle that will give this sine?”

EngageNY Mathematics Algebra II > Module 2 > Topic A > Lesson 4 — Use questions about Quadrant I only

Lesson 13

Lesson 15

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

References

Problem 3

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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