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Math  /  Trigonometry

Question1. Conaider two angles, μ\mu and φ\varphi such that  maider two angles, μ and φ such that π2≤μ≤π2 and sin⁡μ=2522≤φ≤1π2 and sin⁡φ=−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

STEP 1

1. The angles μ\mu and φ\varphi are given with specific ranges and sine values.
2. We need to determine the cosine of the sum of these angles.
3. The trigonometric identities and properties of angles will be used.

STEP 2

1. Interpret the given conditions for μ\mu and φ\varphi.
2. Sketch μ\mu and φ\varphi on separate Cartesian planes.
3. Use the given sine values to find the corresponding cosine values.
4. Apply the cosine addition formula to find cos⁡(μ+φ)\cos(\mu + \varphi).

STEP 3

Interpret the given conditions for μ\mu and φ\varphi:
- For μ\mu, the range is π2≤μ≤π2\frac{\pi}{2} \leq \mu \leq \frac{\pi}{2}, which seems to be a typo. Assuming it means π2≤μ≤π\frac{\pi}{2} \leq \mu \leq \pi, μ\mu is in the second quadrant. - sin⁡μ=25\sin \mu = \frac{2}{5}.
- For φ\varphi, the range is 2π2≤φ≤3π2\frac{2\pi}{2} \leq \varphi \leq \frac{3\pi}{2}, which means φ\varphi is in the third quadrant. - sin⁡φ=−1213\sin \varphi = -\frac{12}{13}.

STEP 4

Sketch μ\mu and φ\varphi on separate Cartesian planes:
- For μ\mu, draw the angle in the second quadrant with a vertical line representing sin⁡μ=25\sin \mu = \frac{2}{5}. - For φ\varphi, draw the angle in the third quadrant with a vertical line representing sin⁡φ=−1213\sin \varphi = -\frac{12}{13}.

STEP 5

Use the given sine values to find the corresponding cosine values:
- For μ\mu, use the identity sin⁡2μ+cos⁡2μ=1\sin^2 \mu + \cos^2 \mu = 1:
$ \cos^2 \mu = 1 - \left(\frac{2}{5}\right)^2 = 1 - \frac{4}{25} = \frac{21}{25} \]
Since μ\mu is in the second quadrant, cos⁡μ\cos \mu is negative:
$ \cos \mu = -\sqrt{\frac{21}{25}} = -\frac{\sqrt{21}}{5} \]
- For φ\varphi, use the identity sin⁡2φ+cos⁡2φ=1\sin^2 \varphi + \cos^2 \varphi = 1:
$ \cos^2 \varphi = 1 - \left(-\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169} \]
Since φ\varphi is in the third quadrant, cos⁡φ\cos \varphi is negative:
$ \cos \varphi = -\sqrt{\frac{25}{169}} = -\frac{5}{13} \]

STEP 6

Apply the cosine addition formula to find cos⁡(μ+φ)\cos(\mu + \varphi):
The cosine addition formula is:
cos⁡(μ+φ)=cos⁡μcos⁡φ−sin⁡μsin⁡φ\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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