TR.UC: Unit Circle

TR.UC.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Radians

TR.UC.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.

Cosine Function
Sine Function
Tangent Function

TR.UC.3: Use special triangles to determine the values of sine, cosine, and tangent for pi/3, pi/4, and pi/6. Apply special right triangles to the unit circle and use them to express the values of sine, cosine, and tangent for x, pi ± x, and 2pi ± x in terms of their values for x, where x is any real number.

Cosine Function
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

TR.T: Triangles

TR.T.1: Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles and the coordinates on the unit circle.

Cosine Function
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

TR.T.2: Solve real-world problems with and without technology that can be modeled using right triangles, including problems that can be modeled using trigonometric ratios. Interpret the solutions and determine whether the solutions are reasonable.

Sine, Cosine, and Tangent Ratios

TR.T.3: Explain and use the relationship between the sine and cosine of complementary angles.

Cosine Function
Sine Function

TR.PF: Periodic Functions

TR.PF.1: Graph trigonometric functions with and without technology. Use the graphs to model and analyze periodic phenomena, stating amplitude, period, frequency, phase shift, and midline (vertical shift).

Cosine Function
Sine Function
Tangent Function
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions

TR.PF.3: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Cosine Function
Sine Function
Tangent Function

TR.PF.4: Construct the inverse trigonometric functions of sine, cosine, and tangent by restricting the domain.

Tangent Function

TR.ID: Identities

TR.ID.1: Prove the Pythagorean identity sin²(x) + cos²(x) = 1 and use it to find trigonometric ratios, given sin(x), cos(x), or tan(x), and the quadrant of the angle.

Cosine Function
Sine Function
Sine, Cosine, and Tangent Ratios

TR.ID.2: Verify trigonometric identities and simplify expressions using trigonometric identities.

Cosine Function
Simplifying Trigonometric Expressions
Sine Function
Sum and Difference Identities for Sine and Cosine

TR.ID.3: Prove the addition and subtraction identities for sine, cosine, and tangent. Use the identities to solve problems.

Sum and Difference Identities for Sine and Cosine

TR.PC: Polar Coordinates and Complex Numbers

TR.PC.1: Understand and use complex numbers, including real and imaginary numbers, on the complex plane in rectangular and polar form, and explain why the rectangular and polar forms of a given complex number represent the same number.

Points in the Complex Plane

TR.PC.3: Define polar coordinates and relate polar coordinates to Cartesian coordinates.

Points in the Complex Plane

TR.PC.4: Translate equations from rectangular coordinates to polar coordinates and from polar coordinates to rectangular coordinates. Graph equations in the polar coordinate plane.

Points in the Complex Plane

TR.V: Vectors

TR.V.1: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||).

Adding Vectors
Vectors

TR.V.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Adding Vectors

TR.V.3: Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Adding Vectors
Vectors

TR.V.7: Solve problems involving velocity and other quantities that can be represented by vectors.

Adding Vectors
Vectors