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2022 G-RT-1-3

Page history last edited by Brigit Minden 10 months, 4 weeks ago

G.RT.1.3


G.RT.1.3 Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions to find the measure of an acute angle in right triangles.
 


In a Nutshell

For right triangles, students will write the appropriate sine, cosine, and tangent ratios for a specific acute angle. Students will find the degree measure for a specified acute angle using inverse trigonometric functions.

 

Student Actions

Teacher Actions

  • Develop accurate and appropriate procedural fluency by attending to accuracy when writing the ratio of side lengths to determine the sine, cosine, and tangent of an acute angle of a right triangle.

  • Develop mathematical reasoning by developing arguments and/or counterarguments to explain the relationship between the trigonometric ratios and similar right triangles. 

  • Develop strategies for problem-solving by applying various techniques such as drawing the diagram, identifying the acute angle in use, as well as labeling the sides of the triangle as adjacent, opposite, and hypotenuse, and using those labels to accurately find the correct trigonometric ratio. 

  • Build procedural fluency from conceptual understanding by providing students with opportunities that allow them to connect trigonometric ratios to properties of similar right triangles, using those connections to apply the appropriate trigonometric ratio needed to find the solution for the situation. 

  • Use and connect representations of various models of situations that incorporate the use of right triangles that allow students to explore and analyze strategies for determining appropriate trigonometric ratios to use when finding the value of a missing angle. 

  • Pose purposeful questions which allow students to develop a deeper understanding of the relationships between right triangle trigonometric ratios and acute angle measures of various right triangle models. 

Key Understandings

Misconceptions 

  • Three basic trigonometric ratios are used to represent acute angle measures of right triangles:

    • Sine of an acute angle in a right triangle is the ratio of the leg opposite the acute angle over the hypotenuse.

    • Cosine of an acute angle in a right triangle is the ratio of the leg adjacent to the acute angle over the hypotenuse. 

    • Tangent of an acute angle in a right triangle is the ratio of the leg opposite the acute angle over the adjacent leg. 

  • Similar right triangles are integral in the study of right triangle trigonometry. For example, given a triangle with sides of 3 in, 4 in, 5 in, and another triangle with sides of 9 in, 12 in, and 15 in the acute angles in the two triangles will be the same and can be verified using inverse trigonometric functions.

  • Solving a right triangle when two side lengths and an angle measure are known involves using Pythagorean Theorem to find the third side, then using proportions and inverse trigonometric ratios to find one of the acute angle measures, and finally using complementary angle measures to find the final acute angle measure

  • Inverse trigonometric functions, sin-1, cos-1, tan-1, can be used to find the acute angle measures of a right triangle. 

  • Common ratios of special right triangles are useful when determining the sine, cosine, and tangent of 30,45, and 60-degree angles. 

  • Students may incorrectly identify the opposite and the adjacent sides of right triangles when labeling a diagram to solve a problem.

  • Students may attempt to use a trigonometric ratio for the right angle and label the hypotenuse on the opposite side.

  • Students may incorrectly use the definitions of the trigonometric functions and misapply the ratios.

  • Students may not use the inverse trig functions when trying to find the measure solve for an acute angle of a right triangle.

  • Students misinterpret the connection that trig ratios are based on similarity and may believe that a right triangle with larger side lengths will have larger values for the trig ratios even though it is similar to a right triangle with proportionally smaller side lengths. 

  Knowledge Connections

Prior Knowledge

Leads to 

  • Construct logical arguments to prove triangle similarity (AA, SSS, SAS). (G.2D.1.10) 
  • Interpret the meanings of quantities involving functions and their inverses. (PC.F.2.3)

  • Estimate trigonometric values of any angle. (PC.T.1.6) 

 

OKMath Framework Introduction

Geometry Grade Introduction

 

 

 

 

 

 

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