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I need help with this practice Having trouble solving it The subject is trigonometry

I need help with this practice Having trouble solving it The subject is trigonometry-example-1
User Dassum
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To solve the problem, we will make use of the identity:


\cos (\alpha-\beta)=\cos (\alpha)\cos (\beta)+\sin (\alpha)\sin (\beta)_{}

ANGLE α

The angle lies in the second quadrant. The only positive ratio is the sine.

If we have that:


\tan \alpha=-(12)/(5)

Displaying this on a triangle for ease of working, we have:

Therefore, the length of the hypotenuse will be:


\begin{gathered} x=\sqrt[]{12^2+5^2}=\sqrt[]{144+25}=\sqrt[]{169} \\ x=13 \end{gathered}

Therefore, we have that:


\begin{gathered} \sin \alpha=(12)/(13) \\ \cos \alpha=-(5)/(13) \end{gathered}

ANGLE β

This angle lies in the fourth quadrant. Only the cosine ratio is positive in this quadrant.

We are given in the question:


\cos \beta=(3)/(5)

Displaying this on a triangle for ease of working, we have:

Therefore, using the Pythagorean Triplets, we have that:


y=4

Therefore, we have that:


\sin \beta=-(4)/(5)

SOLVING THE IDENTITY

Applying the identity quoted earlier, we have:


\begin{gathered} \cos (\alpha-\beta)=\cos (\alpha)\cos (\beta)+\sin (\alpha)\sin (\beta)_{} \\ \cos (\alpha-\beta)=(-(5)/(13))((3)/(5))+((12)/(13))(-(4)/(5)) \\ \cos (\alpha-\beta)=-(63)/(65) \end{gathered}

I need help with this practice Having trouble solving it The subject is trigonometry-example-1
I need help with this practice Having trouble solving it The subject is trigonometry-example-2
User Mike Campbell
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