the exact value of cos(α - β) = 33/65
Explanation:tan α = -12/5
where angle α lies in the 2nd quadrant
In the second quadrant, only sine is positive. tan and cos will be negative
cos β = 3/5
where angle β lies in the 4th quadrant
In the 4th quadrant, only cos is positive. tan and sin will be negative
We are to find cos(α - β)
In trigonometry identity:
[tex]\cos \mleft(\alpha-\beta\mright)\text{ = }cos\alpha\text{ cos}\beta\text{ - sin}\alpha\text{ sin}\beta[/tex]we need to find cosα, sinα and sin β
[tex]\begin{gathered} \tan \text{ = opposite/adjacent} \\ \text{opp = 12, adj = 5},\text{ hyp} \\ \text{hyp}^2=(-12)^2+5^2\text{ = 169 } \\ \text{hyp = }\sqrt[]{169} \\ \text{hyp = 13} \\ \\ \cos \alpha\text{ = }\frac{adj}{hyp} \\ \cos \alpha\text{ = }\frac{5}{13} \\ Since\text{ }\cos \text{ is negative in II, }\cos \alpha\text{ = -}\frac{5}{13} \end{gathered}[/tex]Next we will find sinα:
[tex]\begin{gathered} sin\text{ = opp/hyp} \\ \sin \text{ }\alpha\text{ = }\frac{12}{13}\text{ (sine is positive in quadrant II)} \end{gathered}[/tex]Next we wll find sin β:
[tex]\begin{gathered} \sin \text{ = opp/hyp} \\ \cos \beta=\frac{3}{5} \\ \cos \text{ = adj/hyp} \\ \text{adj = 3, hyp = 5} \\ \text{hyp}^2=opp^2+adj^2 \\ 5^2\text{ = }opp^2+\text{ }3^2 \\ \text{opp}^2\text{ = 25 -9} \\ \text{opp = }\sqrt[]{16}\text{ = 4} \end{gathered}[/tex][tex]\begin{gathered} \sin \text{ }\beta\text{ = 4/5 } \\ \text{Because sin is negative in 4th quadrant, sin }\beta\text{ = -4/5 } \end{gathered}[/tex]substitute the values:
[tex]\begin{gathered} \cos (\alpha-\beta)\text{ = }cos\alpha\text{ cos}\beta\text{ - sin}\alpha\text{ sin}\beta \\ \cos (\alpha-\beta)\text{ =}\frac{\text{ }-5}{13}\text{ }\times\frac{3}{5}\text{- }\frac{12}{13}\text{ }\times\text{ }\frac{-4}{5} \\ \cos (\alpha-\beta)\text{ =}\frac{\text{ }-3}{13}\text{ - }\frac{-48}{65} \\ \cos (\alpha-\beta)\text{ =}\frac{\text{ }-3}{13}\text{+ }\frac{48}{65} \end{gathered}[/tex]simplify:
[tex]\begin{gathered} \cos (\alpha-\beta)\text{ = }\frac{-3(5)+48}{65}\text{ } \\ \cos (\alpha-\beta)\text{ = }\frac{-15+48}{65}\text{ } \\ \cos (\alpha-\beta)\text{ = }\frac{33}{65}\text{ } \end{gathered}[/tex]Hence, the exact value of cos(α - β) = 33/65
