* Use the provided sheets. * This notebook should be your best written work. Quality counts in this project. Proper notation and terminology is important. We will follow the order used in class. Anyone in a Trigonometry course (or beyond) should to be able to read and understand your material. * You are encouraged to provide examples for your own benefit when using this in the future. There may not be sufficient space to neatly put them on the page itself. Use the back or other paper. These items need to be included in the notebook. Follow the order listed. 1. Definitions of sine and cosine based on the unit circle. Definitions of tangent, cotangent, secant, and cosecant based on sine and cosine.. A completed Unit Circle, Trigonometric Table, and master list of all key trigonometric identities. 3. Complete and accurate graphs of all six trigonometric functions. (Let π = 3 squares.) (List domain, range, and period for all graphs and the amplitude on the applicable graphs.) 4. Complete and accurate graphs of the two types of inverses for each trigonometric function. (example: x = siny and y = arcsinx ) The two types of inverses will be on the same graph together but should be clearly labeled. List the Domain and Range for both types of inverses. (Let π = 3 squares.) 5. Prove the common identities using the order from class. (The list will be provided to you.) 6. Right triangle definitions of the six trigonometric functions (You must include a visual/picture similar triangle connection between the unit circle definitions of sine, cosine, tangent and SohCahToa.) 7. Derive the formulas for area of a non right triangle. (Include the related diagram.) 8. Prove the laws of sines and cosines. (Include the related diagram.) 1
1. Definitions of sine and cosine based on the unit circle. Definitions of tangent, cotangent, secant, and cosecant based on sine and cosine.
. Unit Circle. For each key point on the unit circle, include the degrees, radians, and coordinates of the associated points. 3
. Complete the Trigonometric Table. [ ] 0,π π, π ( π π, ) [ ] 0,π π, π ( 0,π ) Radians Degrees cosθ sinθ tanθ secθ cscθ cotθ π 0 π π 3π π 4
. Write in each identity. pythagorean identities (all three) cos(-a) = cos A sin(-a) = - sin A cos(a + B) = cos(a B) = cos( π θ ) = sin( π θ ) = sin(a + B) = sin(a B) = tan (A + B) = tan (A B) = cos A = = = sin A = tan A = Power Reducing Identities: cos θ = sin θ = cos( θ )= sin( θ )= tan( θ )= 5
3. Complete and accurate graphs of all six trigonometric functions. (1 sq. = 1 unit. Let π be 3 sq.) (List domain, range, and period for all graphs and the amplitude on the applicable graphs.) y = sinθ y = cscθ D: R: D: R: y = cosθ y = secθ D: R: D: R: y = tanθ y = cotθ D: R: D: R: 6
4. Complete and accurate graphs of the two types of inverses for each trigonometric function. (example: x = siny and y = arcsinx ) The two types of inverses will be on the same graph together but should be clearly labeled. List the Domain and Range for both types of inverses. (1 sq. = 1 unit. Let π be 3 sq.) 7
5. Prove the common identities. (Follow the order from class. Include a picture if * is shown.) *pythagorean identities (all three) *cos(-a) = cos A *sin(-a) = - sin A 8
5. Prove the common identities. (Follow the order from class. Include a picture if * is shown.) * cos( A+ B) = cos Acos B sin Asin B 9
5. Prove the common identities. (Follow the order from class.) cos( A B) = cos Acos B+ sin Asin B cos ( ) π θ = sinθ sin ( ) π θ = cosθ sin( A+ B) = sin Acos B+ cos Asin B sin( A B) = sin Acos B cos Asin B 10
5. Prove the common identities. (Follow the order from class.) tan A+ tan B tan( A+ B) = 1 tan Atan B tan A tan B tan( A B) = 1 + tan Atan B cos( ) cos sin A = A A = cos A 1 = 1 sin A 11
5. Prove the common identities. (Follow the order from class.) sin A= sin Acos A tan A tan A = 1 tan A Power Reducing Identities: = ( + θ) sin θ = 1 ( 1 cos θ) cos θ 1 cos 1 1
5. Prove the common identities. (Follow the order from class.) cos( θ )= sin( θ )= tan θ ( ) = ± 1 cosθ = 1+ cosθ sin A 1+ cos A = 1 cos A sin A 13
6. Right triangle definitions of the six trigonometric functions (You must include a visual/picture similar triangle connection between the unit circle definitions of sine, cosine, tangent and SohCahToa.) 14
7. Derive the formulas for area of a non right triangle. (Include the related diagram.) (Derive two from this diagram and simply list the third case.) 15
8. Prove the laws of sines and cosines. (Include the related diagram.) (Proofs follow from the previous page. Prove only one law of cosines but list all three.) 16