We know that at exactly 22 12 minutes after the hour, the minute hand of the clock is at point P. Several minutes later, it has rotated θ degrees clockwise to point Q. We are given the coordinates of point Q.
(cos - ( θ +45^(∘) ), sin - (θ +45^(∘) ))
We want to write these coordinates in terms of cos θ and sin θ. To do so, we can rewrite the coordinates by using the Negative Angle Identities.
Negative Angle Identities
sin (- θ) =& - sin θ
cos (- θ) =& cos θ
We can apply these identities to the coordinates of Q.
(cos - ( θ +45^(∘) ), sin - (θ +45^(∘) ))
⇓
(cos ( θ +45^(∘) ), - sin (θ +45^(∘) ))
Next, we will apply the Angle Sum Identity for cosine to find the x-coordinate of Q.
cos ( A+ B)=cos A cos B - sin A sin B
⇓
cos ( θ +45^(∘) ) = cos θ cos 45^(∘)- sin θ sin 45^(∘)
Now, let's recall some trigonometric values for special angles.
Trigonometric Values for Special Angles
Sine
Cosine
sin 30^(∘)=1/2
cos 30^(∘)=sqrt(3)/2
sin 45^(∘)=sqrt(2)/2
cos 45^(∘)=sqrt(2)/2
sin 60^(∘)=sqrt(3)/2
cos 60^(∘)=1/2
We will use the values for 45^(∘) to simplify the previous expression.
We can follow a similar process to find the y-coordinate. In this case, we can recall the Angle Sum Identity for sine.
sin ( A+ B)=sin A cos B + cos A sin B
Now, we will apply this identity to y-coordinate. Then, we will use the trigonometric values for 45^(∘). Let's do it!
