- Express \(285^{\circ}\) as the sum of standard angles, \(240^{\circ} + 45^{\circ}\).
- Apply the cosine addition formula: \(\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)\).
- Use the known values \(\cos(240^{\circ}) = -1/2\), \(\sin(240^{\circ}) = -\sqrt{3}/2\), \(\cos(45^{\circ}) = \sqrt{2}/2\), and \(\sin(45^{\circ}) = \sqrt{2}/2\).
- Simplify the resulting expression to get \(\cos(285^{\circ}) = \boxed{\frac{\sqrt{6}-\sqrt{2}}{4}}\).
Explanation
- Problem Analysis and Strategy
To find the exact value of \(\cos(285^{\circ})\), we observe that the angle \(285^{\circ}\) is not one of the standard angles (like \(30^{\circ}, 45^{\circ}, 60^{\circ}\)) for which we directly know the cosine and sine values. However, we can express it as the sum or difference of two standard angles. A convenient choice is \(285^{\circ} = 240^{\circ} + 45^{\circ}\). Other possibilities exist, such as \(315^{\circ} - 30^{\circ}\). We will proceed using the addition formula for cosine, which states that \(\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\).
- Identifying Trigonometric Values
We use the values for the angles \(240^{\circ}\) and \(45^{\circ}\):
- For \(240^{\circ}\), which is in the third quadrant:
\(\cos(240^{\circ}) = -\frac{1}{2}\)
\(\sin(240^{\circ}) = -\frac{\sqrt{3}}{2}\)
- For \(45^{\circ}\), which is in the first quadrant:
\(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}\)
\(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\)
- Calculation and Simplification
Substituting these values into the cosine addition formula \(\cos(240^{\circ} + 45^{\circ}) = \cos(240^{\circ})\cos(45^{\circ}) - \sin(240^{\circ})\sin(45^{\circ})\):
\[\cos(285^{\circ}) = \left(-\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(-\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\]
Simplifying the terms:
\[\cos(285^{\circ}) = -\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\]
\[\cos(285^{\circ}) = \frac{\sqrt{6} - \sqrt{2}}{4}\]
- Final Conclusion
The exact value of \(\cos(285^{\circ})\) is \(\frac{\sqrt{6} - \sqrt{2}}{4}\). This value can also be verified using the difference formula \(\cos(315^{\circ} - 30^{\circ}) = \cos(315^{\circ})\cos(30^{\circ}) + \sin(315^{\circ})\sin(30^{\circ}) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\), which yields the same result. Thus, \(\boxed{\frac{\sqrt{6}-\sqrt{2}}{4}}\).
Examples
Trigonometric functions are essential in fields like physics and engineering, particularly when dealing with waves, rotation, or oscillation. For example, if you are modeling the path of a rotating robotic arm or the cycle of an alternating current, you often need to calculate the precise coordinates or values at specific angles. Knowing how to decompose complex angles using identities allows you to determine exact position values without relying on approximations, ensuring precision in design and analysis.