Math  /  Trigonometry

Question1. Conaider two angles, μ\mu and φ\varphi such that  maider two angles, μ and φ such that π2μπ2 and sinμ=2522φ1π2 and sinφ=1213\begin{array}{l} \text { maider two angles, } \mu \text { and } \varphi \text { such that } \\ \frac{\pi}{2} \leq \mu \leq \frac{\pi}{2} \text { and } \sin \mu=\frac{2}{5} \quad \frac{2}{2} \leq \varphi \leq \frac{1 \pi}{2} \text { and } \sin \varphi=-\frac{12}{13} \end{array} a. Skitch μ\mu and φ\varphi on separate Cartesian planes. b. Determine the eact value of cos(μ+φ)\cos (\mu+\varphi).

Studdy Solution
Apply the cosine addition formula to find cos(μ+φ)\cos(\mu + \varphi):
The cosine addition formula is:
cos(μ+φ)=cosμcosφsinμsinφ\cos(\mu + \varphi) = \cos \mu \cos \varphi - \sin \mu \sin \varphi
Substitute the values:
cos(μ+φ)=(215)(513)(25)(1213)\cos(\mu + \varphi) = \left(-\frac{\sqrt{21}}{5}\right)\left(-\frac{5}{13}\right) - \left(\frac{2}{5}\right)\left(-\frac{12}{13}\right)
Simplify:
cos(μ+φ)=2113+2465\cos(\mu + \varphi) = \frac{\sqrt{21}}{13} + \frac{24}{65}
Convert 2465\frac{24}{65} to a common denominator with 2113\frac{\sqrt{21}}{13}:
cos(μ+φ)=2113+2465=52165+2465\cos(\mu + \varphi) = \frac{\sqrt{21}}{13} + \frac{24}{65} = \frac{5\sqrt{21}}{65} + \frac{24}{65}
Combine:
cos(μ+φ)=521+2465\cos(\mu + \varphi) = \frac{5\sqrt{21} + 24}{65}
The exact value of cos(μ+φ)\cos(\mu + \varphi) is 521+2465\boxed{\frac{5\sqrt{21} + 24}{65}}.

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