Understands and applies trigonometric relationships
Trigonometry deals with the relationships among the sides and angles of triangles. It provides methods of measuring these sides and angles. Trigonometry is used in solving problems involving triangles, circular motion, and periodic relationships. A fundamental trigonometric relationship in a right triangle is sine of an angle, which is the ratio of the length of the side opposite the angle to the length of the hypotenuse (i.e., the side opposite the right angle).
Students should study trigonometry with respect right triangles, general triangles, circles, and periodic relationships. The relationships between the sides and the angles of triangles, especially right triangles, and the calculations based on them are a focus of triangle trigonometry. Included in the study of right triangle trigonometry are the trigonometric ratios, the Pythagorean theorem and its converse, and the two special-case triangles (30-60-90 and 45-45-90). Trigonometry should be extended beyond right triangles to general triangles using the Law of Sines and Law of Cosines.
Examining right triangles in relation to the unit circle extends analysis to general periodic relationships. Degree and radian measure should be studied. The analysis of trigonometric functions should include: domain and range, period, amplitude, and vertical and horizontal shifts. Further, students should be able to recognize and model relevant periodic phenomenon with trigonometric functions.
Students should use trigonometry to solve problems. Students should reason about, reason with, and apply selected fundamental trigonometric relationships, such as sin2x + cos2x = 1, tanx = sinx/cosx, and cosx = sin (90 - x). In addition, students should be able to solve simple trigonometric equations (e.g., sinx = 0.7).