Lesson:
Sin(u+v)=sinucosv+cosusinv
Sin(u-v)=sinucosv-cosusinv
cos(u+v)=cosucosv-sinusinv
cos(u-v)=cosucosv+sinusinv
tan(u+v)=(tanu+tanv)/(1-tanutanv)
tan(u-v)=(tanu-tanv)/(1+tanutanv)
Easy example problems:
Find the exact value of cos 90°
Use the unit circle to find the degrees that will either add up to or subtract to 90°.
cos(u+v)=cos30°+60°=cos30°cos60°-sin30°sin60°
convert on unit circle- cos=x sin=y
√3/2(1/2) - 1/2 √3/2
This is your simplified answer.
How about a harder one?
Find the exact value of sine, cosine and tangent of 285°
Using the unit circle see what degrees can add up to or subtract to 285°.
cos(240° + 45°)= cos240° cos45° - sin 240° sin45°
= -√2/2(√2/2) - -√2/2(√2/2)
sin(330°- 45°) = sin 330° cos45°- sin330° sin45°
=-1/2(√2/2) - √3/2(√2/2)
tan(240°+45°) = tan 240° + tan 45°/ 1 - tan240° tan45°
=(-(√2/2)/(-√2/2) + (√2/2)/(√2/2)) / 1- (-√2/2)/(-√2/2) ((√2/2) / (√2/2))
Now real world application problems
1.
A street sign hangs from a pole. When the wind blows, it moves the sign vertically from its original position which is shown by
y=1/3Sin2t=1/4Cos2t
y=distance from equilibrium in feet
t=the time in seconds
use this identity-
a sin B of theta+b cos B of theta=√a^2+b^2Sin(B of theta +C)
C here is the arctan(b/a), a is greater than 0
If you use this to guide you, you can make the original equation into the following
y=√a^2+b^2sin(Bt+C)
Use a calculator to divide and multiply the numbers to end up with
y=5/12sin(2t=.6435)
2. Imagine you are walking on a path that immitates the unit circle. Use the sum and difference formula to calculate the Tangent if one of the degrees is 59 and one is 38. First find the equation appropriate to use...tan(u + v) = tan(u) + tan(v) 1 - tan(u)tan(v)
With u = 59° and v = 38°, we have...
tan(59 + 38) = tan(59) + tan(38) 1 - tan(59)tan(38)
On the unit circle, convert these by finding the exact values of these numbers
tan(59 + 38) = 1.6643 + 0.7813 1 - (1.6643)(0.7813)
tan(59 + 38) = 2.4456 1 - 1.30031759
tan(59 + 38) = 2.4456 -0.30031759
By pluging in the numbers we come to find that...
tan(59 + 38) = -8.14337914739
Video to help with Sum/Difference
http://www.youtube.com/watch?v=zdTbFVOJIpA
Find the exact value of cos 90°
Use the unit circle to find the degrees that will either add up to or subtract to 90°.
cos(u+v)=cos30°+60°=cos30°cos60°-sin30°sin60°
convert on unit circle- cos=x sin=y
√3/2(1/2) - 1/2 √3/2
This is your simplified answer.
How about a harder one?
Find the exact value of sine, cosine and tangent of 285°
Using the unit circle see what degrees can add up to or subtract to 285°.
cos(240° + 45°)= cos240° cos45° - sin 240° sin45°
= -√2/2(√2/2) - -√2/2(√2/2)
sin(330°- 45°) = sin 330° cos45°- sin330° sin45°
=-1/2(√2/2) - √3/2(√2/2)
tan(240°+45°) = tan 240° + tan 45°/ 1 - tan240° tan45°
=(-(√2/2)/(-√2/2) + (√2/2)/(√2/2)) / 1- (-√2/2)/(-√2/2) ((√2/2) / (√2/2))
Now real world application problems
1.
A street sign hangs from a pole. When the wind blows, it moves the sign vertically from its original position which is shown by
y=1/3Sin2t=1/4Cos2t
y=distance from equilibrium in feet
t=the time in seconds
use this identity-
a sin B of theta+b cos B of theta=√a^2+b^2Sin(B of theta +C)
C here is the arctan(b/a), a is greater than 0
If you use this to guide you, you can make the original equation into the following
y=√a^2+b^2sin(Bt+C)
Use a calculator to divide and multiply the numbers to end up with
y=5/12sin(2t=.6435)
2. Imagine you are walking on a path that immitates the unit circle. Use the sum and difference formula to calculate the Tangent if one of the degrees is 59 and one is 38. First find the equation appropriate to use...tan(u + v) = tan(u) + tan(v) 1 - tan(u)tan(v)
With u = 59° and v = 38°, we have...
tan(59 + 38) = tan(59) + tan(38) 1 - tan(59)tan(38)
On the unit circle, convert these by finding the exact values of these numbers
tan(59 + 38) = 1.6643 + 0.7813 1 - (1.6643)(0.7813)
tan(59 + 38) = 2.4456 1 - 1.30031759
tan(59 + 38) = 2.4456 -0.30031759
By pluging in the numbers we come to find that...
tan(59 + 38) = -8.14337914739
Video to help with Sum/Difference
http://www.youtube.com/watch?v=zdTbFVOJIpA
