Application for Addition Formulae

Application for Addition Formulae: Part 2

by J Drauna

​RECAP on Part 1

​To write the sum of a Sine/Cosine as a function using the Addition Formulae; You need to :

1. ​Find the Modulus

​

​

1. ​Find Alpha

Multiple Choice

$$y\ =\ 2\ \sin\ \theta+\cos\theta$$  

What is the modulus (R) of the above expression?

• $$2$$  

• $$5$$  

• $$\sqrt[]{5}$$  

• $$1$$  

Poll

How difficult was it for you to find the modulus in the earlier question?

Multiple Choice

Which of the following is the correct expression for: $$R\ \cos\ \left(\theta+\alpha\right)$$

 

• $$\sin\ \theta\ \ \sin\alpha\ +\ \cos\theta\ \sin\alpha$$  

• $$\sin\ \theta\cos\alpha+\cos\theta\sin\alpha$$  

• $$\cos\theta\cos\alpha-\sin\theta\sin\alpha$$  

• $$\cos\theta\cos\alpha+\sin\theta\sin\alpha$$  

Poll

After having equate both expressions correctly; it looks something like this. $$2\sin\theta+\cos\theta=\sqrt[]{5}\cos\theta\cos\alpha-\sqrt[]{5}\sin\theta\sin\alpha$$  

How confident are you in comparing the sine and cosine of $$\theta$$  ?

Multiple Choice

SO what are the correct comparisons for $$\sin\ \theta\ \$$  and $$\cos\theta$$   for the function $$2\sin\theta+\cos\theta=\sqrt[]{5}\cos\theta\cos\alpha-\sqrt[]{5}\ \ \sin\theta\sin\alpha$$  ?

• $$2\sin\theta=\sqrt[]{5}\cos\theta\cos\alpha$$     and $$\cos\ \theta=-\sqrt[]{5}\sin\theta\sin\alpha$$  

• $$2\ \sin\ \theta=\sqrt[]{5}\sin\theta\sin\alpha$$   and $$\cos\theta=-\sqrt[]{5}\cos\theta\cos\alpha$$  

• $$2\sin\theta=-\sqrt[]{5}\sin\ \theta\sin\alpha$$    

  and 

    $$\cos\theta=\sqrt[]{5}\cos\theta\cos\alpha$$  

​Solving ALPHA

​Identify in what Quadrant would they share in common?

​

​

Multiple Choice

With sine being negative ;  $$\sin\ \alpha=\frac{2}{-\sqrt[]{5}}$$   and cosine being positive;  $$\cos\alpha=\frac{1}{\sqrt[]{5}}$$  , which Quadrant would they  both share ALPHA?

• Quadrant 1

• Quadrant 2

• Quadrant 3

• Quadrant 4

​TRUE VALUE OF ALPHA

Since both have in Quadrant 4;