P is in the Quadrant II.
We have: angle t = (pi/2 + angle a).∠t=(π2+∠a).
Angle a is defined by tan a = 2/3tana=23 --> sin a = 2/sqrt(4 + 9) = 2/sqrt13sina=2√4+9=2√13 -->
cos^2 a = 1 - sin^2 a = 1 - 4/13 = 9/13cos2a=1−sin2a=1−413=913 --> cos a = 3/sqrt13cosa=3√13.
Find the trig functions of t.
sin t = sin (a + pi/2) = cos a = 3/sqrt13sint=sin(a+π2)=cosa=3√13
cos t = cos (pi/2 + a) = - sin a = - 2/sqrt13cost=cos(π2+a)=−sina=−2√13
sin 2t = 2sin t.cos t = - (2/sqrt13)(3/sqrt13) = - 6/13sin2t=2sint.cost=−(2√13)(3√13)=−613
cos^2 2t = 1 - sin^2 t = 1 - 36/169 = 133/169cos22t=1−sin2t=1−36169=133169
cos 2t = - sqrt133/13cos2t=−√13313. (2t is in Quadrant III)
sin (t - 150) = sin t.cos 150 - sin 150.cos t.
= - (3/sqrt13)(-sqrt3/2) - (1/2)(-2/sqrt13) ==−(3√13)(−√32)−(12)(−2√13)=
= - ((3sqrt3)/(2sqrt13)) + (2/(2sqrt13)) = (2 - 3sqrt3)/(2sqrt13)=−(3√32√13)+(22√13)=2−3√32√13
cos (225 - t) = cos 225.cos t + sin 225.sin t =
cos 225 = -cos 45 = -sqrt2/2cos225=−cos45=−√22 ; and sin 225 = -sin 45 = -sqrt2/2sin225=−sin45=−√22
