Find the exact value of [tex]$\cos (\alpha+\beta)$[/tex], given [tex]$\sin \alpha=\frac{2}{5}$[/tex] for [tex]$\alpha$[/tex] in Quadrant II and [tex]$\cos \beta=\frac{1}{3}$[/tex] for [tex]$\beta$[/tex] in Quadrant IV.
A. [tex]$\frac{8 \sqrt{2}-\sqrt{15}}{15}$[/tex]
B. [tex]$\frac{8 \sqrt{2}+\sqrt{15}}{15}$[/tex]
C. [tex]$\frac{4 \sqrt{2}+\sqrt{21}}{15}$[/tex]
D. [tex]$\frac{4 \sqrt{2}-\sqrt{21}}{15}$[/tex]
Answer (1)
Find cos ( α ) using the Pythagorean identity and the quadrant information: cos α = − 5 21 .
Find sin ( β ) using the Pythagorean identity and the quadrant information: sin β = − 3 2 2 .
Apply the cosine addition formula: cos ( α + β ) = cos α cos β − sin α sin β .
Substitute the values and simplify: cos ( α + β ) = 15 4 2 − 21 .
Explanation
Problem Setup We are given that sin α = 5 2 and α is in Quadrant II. We are also given that cos β = 3 1 and β is in Quadrant IV. Our goal is to find the exact value of cos ( α + β ) .
Cosine Sum Identity Recall the trigonometric identity for the cosine of a sum: cos ( α + β ) = cos α cos β − sin α sin β
Finding cos α We need to find cos α and sin β . Since sin 2 α + cos 2 α = 1 , we have cos 2 α = 1 − sin 2 α = 1 − ( 5 2 ) 2 = 1 − 25 4 = 25 21 Since α is in Quadrant II, cos α is negative. Thus, cos α = − 25 21 = − 5 21
Finding sin β Similarly, since sin 2 β + cos 2 β = 1 , we have sin 2 β = 1 − cos 2 β = 1 − ( 3 1 ) 2 = 1 − 9 1 = 9 8 Since β is in Quadrant IV, sin β is negative. Thus, sin β = − 9 8 = − 3 8 = − 3 2 2
Applying the Cosine Sum Formula Now we can substitute the values of sin α , cos α , sin β , and cos β into the cosine addition formula: cos ( α + β ) = cos α cos β − sin α sin β = ( − 5 21 ) ( 3 1 ) − ( 5 2 ) ( − 3 2 2 )
Final Calculation cos ( α + β ) = − 15 21 + 15 4 2 = 15 4 2 − 21
Final Answer Therefore, the exact value of cos ( α + β ) is 15 4 2 − 21 .
Examples
In physics, when analyzing wave interference, you often need to calculate the cosine of the sum of two angles. For example, if two waves with phases α and β interfere, the resulting wave's amplitude depends on cos ( α + β ) . Knowing sin α and cos β allows you to determine the exact value of cos ( α + β ) , which helps predict the intensity of the resulting wave. This is crucial in fields like optics and acoustics for designing systems that either enhance or cancel out wave interference.
