Download Math 2 Unit 5: Right Triangles and Trigonometry

Math 2 Unit 7: Right Triangles and Trigonometry
Approximate Time Frame: 3 – 4 Weeks
Connections to Previous Learning:
In prior units, students studied the definition of similarity, and worked with proportions of corresponding sides and congruencies of corresponding angles. This
unit will build on these ideas to specifically include the side ratios of right triangles and the definitions of trigonometric ratios for acute angles. Further, the
Pythagorean Theorem (Grade 8) and its proof via similarity (previous unit) will be utilized in solving right triangles.
Focus of this Unit:
Students will use their knowledge of similarity to write the ratio of sides of right triangles as functions of its acute angles, defining the trigonometric ratios. They
will use these trigonometric ratios and the Pythagorean Theorem to solve right triangles and to solve application problems modeled by right triangles.
Connections to Subsequent Learning:
Students will connect their knowledge of right triangle trigonometric ratios to the unit circle and graphic representations of the trigonometric functions.
Desired Outcomes
Standard(s):
Define trigonometric ratios and solve problems involving right triangles.
 G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angle in the triangle, leading to definitions of trigonometric ratios for
acute angles.
 G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
 G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Transfer:
Students will use concepts and skills to find missing measurements of right triangles in various real-world situations using Pythagorean Theorem, special right
triangles, and right triangle trigonometry.
Ex. A fire truck pulls up to a burning building with a person trapped at the 4th floor window which is 45 ft off the ground. The truck is parked 9 ft away from the
building to give the firefighters and police room to work on the ground. The base of the ladder is 5 ft off the ground already because it is sitting up on the truck.
If it takes 2 seconds to extend the ladder one foot, how long will it take the firefighters to extend the ladder to the window?
Ex. A tipping platform is a ramp used to unload trucks. How high is the end of an 80 foot ramp when it is tipped by a 30° angle? By a 45° angle?
Ex. A digital camera with a panoramic lens is described having a view with an angle of elevation of 38 o. If the camera is on a 3 ft tripod aimed
directly at a 124 ft tall statue, how far from the statue should you place the tripod to see very top of the statue through the lens?
3/6/2014 12:14:31 AM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 1
Additional Standards = Approximately 10%
Math 2 Unit 7: Right Triangles and Trigonometry
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 explicit vocabulary instruction with regard to the study of angles in trigonometry.
 tactile and virtual tools to study the relationships between sides and angles in right triangles.
Understandings: Students will understand that …
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The ratios of the sides of right triangles are functions of the acute angles of the triangle.
The sine of an acute angle in a right triangle is equal to the cosine of that angle's complement (and vice versa).
The Pythagorean Theorem applies only to right triangles.
Essential Questions:
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How does similarity give rise to the trigonometric ratios?
How do the trigonometric ratios of complementary angles relate to one another?
How can the Pythagorean Theorem be used to solve problems involving triangles?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students will solve problems in context that involve right triangles.
*2. Reason abstractly and quantitatively. Students reason about the ratios used to represent relationships between sides and angles.
3.
4.
*5.
*6.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically. Students will use calculators to evaluate trigonometric values and to solve for sides of a right triangle.
Attend to precision. While solving problems involving right triangles, students will attend to the precision of their answers. They will use appropriate
language to describe their measurements and calculations.
*7. Look for and make use of structure. Students will set appropriate ratios of the sides of right triangles equal to the sine, cosine, or tangent of an angle.
Also, students will recognize when it is appropriate to use the Pythagorean Theorem.
*8. Look for and express regularity in repeated reasoning. Students will recognize that trigonometric ratios arise from the ratio of sides of similar right
triangles.
3/6/2014 12:14:31 AM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 2
Additional Standards = Approximately 10%
Math 2 Unit 7: Right Triangles and Trigonometry
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Students should already be able to:
 Use the Pythagorean Theorem to solve for a missing side of a
right triangle with integer-valued sides.
3/6/2014 12:14:31 AM
Priority Standards = Approximately 70%
Some students may be ready to:
 See the complementary angle trigonometric identities as horizontal
transformations.
 Define the reciprocal trigonometric functions.
 Define the inverse trigonometric functions.
 Derive the formula for the area of a triangle that uses two sides and sine of the
included angle between those sides. (G.SRT.9+ )
 Prove the Law of Sines and Law of Cosines (G.SRT.10+ )
 Solve for missing sides in non-right triangles, using the Law of Sines and Law of
Cosines. (G.SRT.10+, G.SRT.11+ )
 Recognize and use the slope of a line as the tangent of its angle of elevation.
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 3
Additional Standards = Approximately 10%
Math 2 Unit 7: Right Triangles and Trigonometry
Knowledge: Students will know…
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Skills: Students will be able to …
The trigonometric function definitions of sine, cosine, and
tangent as ratios of the sides of a right triangle.
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Use the trigonometric ratios and knowledge of right triangles to determine the
sine, cosine, and tangent values of 30º, 45º, and 60º without the assistance of
technology.
Apply the Pythagorean Theorem to problems involving right triangles.
Solve for the angles in a right triangle, given at least two sides.
Solve for the missing sides of a right triangle, given either two sides or one
acute angle and one side.
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Hypotenuse
Opposite side
Adjacent side
Sine
Cosine
Tangent
Pythagorean triple
"Solve" a triangle
Complementary angles
Angle of elevation
Angle of depression
3/6/2014 12:14:31 AM
Priority Standards = Approximately 70%
Adapted from UbD framework
Supporting Standards = Approximately 20%
Page 4
Additional Standards = Approximately 10%