Discovering sum & Difference identities © Teresa Scar Fuston 2019
Use the unit circle to find the following: sin(60°) = sin(150°)= sin(210°)= sin(60°+150°)= sin(60°) + sin(150°)= Is sin(60°+150°) = sin(60°) + sin(150°) ? NO! © Teresa Scar Fuston 2019
So ... If sin(a + b) ≠ sin(a) + sin(b), what is ... sin(a + b) = cos(a + b) = We are so glad you asked! :) © Teresa Scar Fuston 2019
In a right triangle, so so Label the legs in the right triangle. © Teresa Scar Fuston 2019
Label the sides of the shaded triangle like we did on the previous slide … notice the angle is now . 1 © Teresa Scar Fuston 2019
That allows us to label the other angle. Since we have parallel lines cut by a transversal, alternate interior angles are congruent. + 1 That allows us to label the other angle. © Teresa Scar Fuston 2019
Now label the sides of this shaded triangle using the angle (+). 1 © Teresa Scar Fuston 2019
The next shaded triangle is different because the hypotenuse is not 1 The next shaded triangle is different because the hypotenuse is not 1! A few slides ago we saw that and . Label the sides of this triangle. 1 + © Teresa Scar Fuston 2019
We can find the marked angle using geometry and algebra. + 1 © Teresa Scar Fuston 2019
Label the sides of the last triangle! + 1 © Teresa Scar Fuston 2019
Compare the vertical sides of the rectangle. They’d have equal lengths Compare the vertical sides of the rectangle. They’d have equal lengths. So … + 1 © Teresa Scar Fuston 2019
Compare the horizontal sides of he rectangle. They’d have equal lengths. So … and, with algebra … + 1 © Teresa Scar Fuston 2019
So ... If sin(a + b) ≠ sin(a) + sin(b), sin(a + b) = sinacosb + cosa sinb cos(a + b) = cosa cosb sina sinb Ta-Da! :) © Teresa Scar Fuston 2019