- Calculate the fundamental values \(\cos(\alpha) = \frac{\sqrt{19}}{5}\) and \(\sin(\beta) = \frac{1}{\sqrt{17}}\).
- Derive the double-angle values: \(\sin(2\alpha) = -\frac{2\sqrt{114}}{25}\), \(\cos(2\alpha) = \frac{13}{25}\), \(\sin(2\beta) = -\frac{8}{17}\), and \(\cos(2\beta) = \frac{15}{17}\).
- Apply sum and difference identities for sine and cosine to find the values:
A\(\sin(\alpha + 2\beta) = \boxed{-\frac{15\sqrt{6} + 8\sqrt{19}}{85}}\)
B\(\cos(2\alpha - \beta) = \boxed{-\frac{(52 + 2\sqrt{114})\sqrt{17}}{425}}\)
C\(\cos(2\alpha - 2\beta) = \boxed{\frac{195 + 16\sqrt{114}}{425}}\)
Explanation
- Determining Fundamental Trigonometric Values
We are given two trigonometric values: \(\sin(\alpha) = -\frac{\sqrt{6}}{5}\) with \(\alpha \in \text{QIV}\) and \(\cos(\beta) = -\frac{4}{\sqrt{17}}\) with \(\beta \in \text{QII}\).
In the fourth quadrant (QIV), the cosine is positive, and in the second quadrant (QII), the sine is positive. Using the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\):
For \(\alpha\): \(\cos(\alpha) = \sqrt{1 - \sin^2(\alpha)} = \sqrt{1 - (-\frac{\sqrt{6}}{5})^2} = \sqrt{1 - \frac{6}{25}} = \sqrt{\frac{19}{25}} = \frac{\sqrt{19}}{5}\).
For \(\beta\): \(\sin(\beta) = \sqrt{1 - \cos^2(\beta)} = \sqrt{1 - (-\frac{4}{\sqrt{17}})^2} = \sqrt{1 - \frac{16}{17}} = \sqrt{\frac{1}{17}} = \frac{1}{\sqrt{17}}\).
- Calculating Double-Angle Identities
We calculate the double-angle values:
\(\sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha) = 2 (-\frac{\sqrt{6}}{5}) (\frac{\sqrt{19}}{5}) = -\frac{2\sqrt{114}}{25}\)
\(\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha) = \frac{19}{25} - \frac{6}{25} = \frac{13}{25}\)
\(\sin(2\beta) = 2 \sin(\beta) \cos(\beta) = 2 (\frac{1}{\sqrt{17}}) (-\frac{4}{\sqrt{17}}) = -\frac{8}{17}\)
\(\cos(2\beta) = \cos^2(\beta) - \sin^2(\beta) = \frac{16}{17} - \frac{1}{17} = \frac{15}{17}\)
- Evaluating the Trigonometric Expressions
AUsing \(\sin(\alpha + 2\beta) = \sin(\alpha)\cos(2\beta) + \cos(\alpha)\sin(2\beta)\):
\(\sin(\alpha + 2\beta) = (-\frac{\sqrt{6}}{5})(\frac{15}{17}) + (\frac{\sqrt{19}}{5})(-\frac{8}{17}) = -\frac{15\sqrt{6}}{85} - \frac{8\sqrt{19}}{85} = -\frac{15\sqrt{6} + 8\sqrt{19}}{85}\).
BUsing \(\cos(2\alpha - \beta) = \cos(2\alpha)\cos(\beta) + \sin(2\alpha)\sin(\beta)\):
\(\cos(2\alpha - \beta) = (\frac{13}{25})(-\frac{4}{\sqrt{17}}) + (-\frac{2\sqrt{114}}{25})(\frac{1}{\sqrt{17}}) = -\frac{52}{25\sqrt{17}} - \frac{2\sqrt{114}}{25\sqrt{17}} = -\frac{52 + 2\sqrt{114}}{25\sqrt{17}} = -\frac{(52 + 2\sqrt{114})\sqrt{17}}{425}\).
CUsing \(\cos(2\alpha - 2\beta) = \cos(2\alpha)\cos(2\beta) + \sin(2\alpha)\sin(2\beta)\):
\(\cos(2\alpha - 2\beta) = (\frac{13}{25})(\frac{15}{17}) + (-\frac{2\sqrt{114}}{25})(-\frac{8}{17}) = \frac{195}{425} + \frac{16\sqrt{114}}{425} = \frac{195 + 16\sqrt{114}}{425}\).
Examples
Trigonometric evaluations like these are foundational in physics and engineering, particularly when analyzing wave interference or oscillating systems. For example, when calculating the resultant amplitude of two interfering waves that arrive at a point with a specific phase difference, engineers use these identities to combine trigonometric functions. Understanding how to break down complex angles helps in modeling real-world physical phenomena such as signal propagation, mechanical vibrations, or even the movement of robotic arms in a coordinate system.