Student Actions

Teacher Actions

• Develop Mathematical Reasoning by applying definitions, postulates, and theorems pertaining to right triangles appropriately when tasked with finding missing side lengths measures using diagrams or mathematical relationships involving the three basic trigonometric functions.

• Develop the ability to communicate mathematically by utilizing mathematical terminology pertaining to right triangle trigonometry with diagrams, graphical models, and both written and verbal arguments with both their teacher and their peers.

• Develop Strategies for Problem-Solving by using and communicating various strategies (drawing and labeling a diagram, using mnemonics to choose the correct trig function, setting up the proper algebraic equation) to find side length measures of right triangles involving trig functions in many contexts. 

• Use and connect mathematical representations by providing students with opportunities to investigate trigonometric ratio properties of right triangles using multiple pictorial representations involving different orientations (making sure that the triangles used in examples are oriented differently).

• Build procedural fluency from conceptual understanding by reviewing and integrating the application of solving proportions to the process of finding missing side length measures of right triangles using trigonometric ratios.

• Implement tasks that promote reasoning and problem-solving by engaging students with tasks involving right triangles that can be solved utilizing different methods, and encouraging students to discuss their strategies with their peers using appropriate terminology.

• Promote productive struggle by providing students with opportunities that require them to identify errors in calculation and communicating an appropriate strategy for utilizing trigonometric ratios to produce the correct measurement(s) of side length in right triangles. 

Key Understandings

Misconceptions 

• Three trigonometric ratios are used to find missing side lengths 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. 

• Problem-solving situations involving trigonometric ratios can be modeled by drawing a right triangle,  labeling the known information, and then using proportions with trigonometric ratios and algebraic techniques to find the missing measures.

• Solving a right triangle when one side length and one of the acute angle measures are known involves using complementary angle measures to find the other acute angle, then using proportions and trigonometric ratios to find one side length, and finally using another trig proportion or Pythagorean Theorem to find the remaining side length. 

• Students may label the sides of the triangle (opposite, adjacent, and hypotenuse) incorrectly in relation to the acute angle based on the given information.

• Students may use incorrect trigonometric ratios.

• Students may set up and/or solve the proportion incorrectly.

• Students may use the inverse trigonometric function when finding the side length of a right triangle. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Use the Pythagorean theorem to find the distance between any two points in a coordinate plane. (PA.GM.1.2)

• Analyze and interpret associations between graphical representations and written scenarios. (A1.A.4.5) 

• Create models for situations involving trigonometry. (PC.T.2.1)

• Use trigonometry to find the area of triangles. (PC.T.2.3)

• Choose and produce an equivalent form of an expression to explain the properties of the quantity represented by the expression. (PC.T.3.2)