Trigonometry addition & substraction id
- 2. Trigonometric Addition Identities For Sine And Cosine
- 3. Proof of the y C A O B a b c P x α β y C O B a c β P x P b C OA α
- 4. example
- 6. 90-A A y xz Proof of the example
- 7. Proof of the
- 8. Quadrant I α Quadrant II 1800 - α Quadrant III 1800+ α Quadrant IV 3600 - α sin + + - - cos + - - + tan + - + - Sin Tan cos sin tan cos
- 10. By using the result for sin2α into our RHS and obtain: (remember: example
- 11. Change sin 70◦cos 150 ◦+cos70 ◦sin150◦into a trigonometric function in a single variable and evaluate it. Answer: This is one side of sum idnetity for sines : sin(α+β)=sin α.cos β+cos α.sin β sin 70◦cos 150 ◦+cos70 ◦sin150◦ = sin (70◦+150◦ ) = sin (220◦) = -sin 220◦ =-sin (220 ◦ - 180 ◦) =-sin 40 ◦ =-.643 180 ◦ <220 ◦ <270 ◦ Quadrant lll =-cos(270-220) =-cos 50 =-.643
- 12. Change sin 60◦cos 45 ◦- cos 60 ◦sin45◦into a trigonometric function in a single variable and evaluate it. Answer: This is part of the difference identity for sines : sin(α-β)=sin α.cos β-cos α.sin β sin 60◦cos 45 ◦- cos 60 ◦sin150◦ = sin (60◦-45◦ ) = sin (15◦) = sin 15 ◦ =.259
- 13. Change cos 85◦cos 15 ◦- sin85◦ sin15◦into a trigonometric function in a single variable and evaluate it. Answer: This is part of the sum identity for cosines : cos(α+β) = cos α.cos β – sin α.sin β cos 85◦cos 15 ◦- sin85◦ sin15◦ = cos (85◦+15◦ ) = cos (100◦) = -cos 100 ◦ =-cos(180 ◦ - 100◦) =-cos 80 ◦ =-.087
- 14. 4 5 3 α If P in the second quadrant and sinP= , find sin2P. Answer: If sinP= , then cosP= sin2P = 2sinP.cosP sin2P= 2 =
- 15. If A is a second quadrant angle, and sinA = what is cos2A ? Answaer: Use pythagorean tripels : (5, 12, 13) If sinA= , then cosA= If we want to use the formula of : Cause cosine in negative in quadrant II →