Exercise 41 Page 949

Use the Angle Sum and Difference Identities.

(5cos θ- 5sqrt(3)sinθ ,5sin θ + 5sqrt(3) cosθ )

Practice makes perfect

We are given a diagram that represents a gear with a radius of 10 centimeters. In this diagram, point A represents a 60^(∘) counterclockwise rotation of point P(10,0). Point B represents a θ-degree rotation of point A.

We know the coordinates of B and we want to write these coordinates in terms of cos θ and sin θ. (10cos ( θ +60^(∘) ), 10sin (θ +60^(∘) ))To do so, we can recall the Angle Sum Identity for cosine. cos ( A+ B)=cos A cos B - sin A sin B We will apply this identity to the x-coordinate.

10cos ( θ +60^(∘) )

Substitute

cos ( θ+ 60^(∘))= cos θ cos 60^(∘)- sin θsin 60^(∘)

10( cos θ cos 60^(∘)- sin θ sin 60^(∘))

Distr

Distribute 10

10cos θ cos 60 ^(∘) - 10sin θ sin 60 ^(∘)

Now, let's remember 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 60 ^(∘) to simplify the previous expression.

10cos θ cos 60 ^(∘) - 10sin θ sin 60 ^(∘)

SubstituteII

cos 60 ^(∘)= 1/2, sin 60 ^(∘)= sqrt(3)/2

10cos θ ( 1/2) - 10sin θ ( sqrt(3)/2)

CommutativePropMult

Commutative Property of Multiplication

10 (1/2 )cos θ - 10(sqrt(3)/2)sin θ

MoveLeftFacToNum

a*b/c= a* b/c

10/2cos θ- 10sqrt(3)/2sinθ

SimpQuot

Simplify quotient

5cos θ- 5sqrt(3)sinθ

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 60^(∘). Let's do it!

10 sin ( θ +60^(∘) )

Substitute

sin ( θ+ 60^(∘))= sin θ cos 60^(∘)+ cos θsin 60^(∘)

10( sin θ cos 60^(∘)+ cos θ sin 60^(∘))