Identities and special values for trigonometric functions
• Prove the Pythagorean Identity $\sin^2\theta+\cos^2\theta=1$ (F-TF.C.8).
• Use the unit circle to prove trigonometric identities and relate them to symmetries of the graphs of sine and cosine (F-TF.A.4(+)).
• Use the Pythagorean Identity to calculate trigonometric ratios (F-TF.C.8).
In this section, which includes some (+) standards, students explore further the consequences of the unit circle definition of sine and cosine. They make a connection between the Pythagorean theorem. Then, they see how the symmetries of the circle give rise to symmetries of the graphs of sine and cosine, and represent these symmetries as identities.
WHAT: Students are guided through a proof of the identity $\sin^2\theta+\cos^2\theta=1$ using the Pythagorean Theorem, and then use the identity to compute a cosine give the sine of an angle in the second quadrant.
WHY: This task draws an important connection between geometry and the study of trigonometric functions. The idea of the sine or the cosine has undergone a considerable evolution, from a ratio attached to a geometric figure to a function modeling periodic behavior, for which the input is often not regarded as an angle any longer. It is easy to forget the geometric origins. This task shows that an identity relating sine and cosine is in fact nothing more than the Pythagorean theorem in disguise. Seeing that structure in the identity can help students remember it (MP.7).
WHAT: Students use the unit circle to explain the trigonometric identities that express the periodicity of the sine and cosine functions, the fact that the sine function is odd, and the fact that the cosine function is even.
WHY: The unit circle provides a unifying view of the sine and cosine functions and can be used to organize the various identities relating the functions. In the previous activity students saw how the unit circle could be used to prove the Pythagorean identity; now they see how the various symmetry properties of the two functions result from the symmetry of the circle itself.
WHAT: Students experiment with the graphs of sine and cosine and discover translations and reflections that take them onto themselves and onto each other, then explain these transformations in terms of trigonometric identities.
WHY: The identities that students studied in the previous activity result in various symmetry properties of the graphs of sine and cosine: the fact that the graphs have a periodic shape, and their reflection properties. The purpose of this task is to help students see those properties and relate them to identities, and to see how other identities can be seen directly from the graphs.
WHAT: Students are given a diagram showing a right triangle with angle $\pi/4$ inside a unit circle, and are asked to use the diagram to find the sine and cosine of $\pi/4$.
WHY: This tasks reminds students of the origins of trigonometric ratios in the study of right triangles, and situates a special right triangle within the unit circle. Students may have already seen a derivation of the sine and cosine of $45^\circ$. Just as they have seen the Pythagorean theorem as the origin of the Pythagorean identity in the activity "F-TF Trigonometric Ratios and the Pythagorean Theorem", now they see the connection between values of sine and cosine at special inputs and their previous work with special triangles. A similar task, Special Triangles 2 (https://www.illustrativemathematics.org/content-standards/tasks/1898), deals with the value of sine and cosine at $\pi/6$.