Evaluate Trigonometric Expression Sin(sin⁻¹ 0 - Cos⁻¹(√3/2))
In the realm of mathematics, particularly in trigonometry, we often encounter complex expressions involving trigonometric functions and their inverses. These expressions require a deep understanding of trigonometric identities, inverse trigonometric functions, and their properties. This article delves into the process of finding the exact value of a specific trigonometric expression, thereby enhancing your grasp of these essential mathematical concepts. Our focus will be on the expression $\sin \left(\sin ^{-1} 0-\cos ^{-1} \frac{\sqrt{3}}{2}\right)$, which combines inverse trigonometric functions within a sine function. To tackle this, we'll methodically break down the problem, leveraging our knowledge of inverse trigonometric values and trigonometric identities to arrive at the precise solution.
Breaking Down the Expression: Inverse Trigonometric Functions
At the heart of the expression $\sin \left(\sin ^{-1} 0-\cos ^{-1} \frac{\sqrt{3}}{2}\right)$ lie the inverse trigonometric functions. To effectively evaluate the entire expression, we need to first decipher the values of these inverse functions individually. The two key components we'll address are $\sin^{-1} 0$ and $\cos^{-1} \frac{\sqrt{3}}{2}$. Understanding these inverse functions is paramount because they essentially ask the reverse question of their regular trigonometric counterparts. For instance, $\\sin^{-1} 0$ asks, "What angle has a sine of 0?" Similarly, $\cos^{-1} \frac{\sqrt{3}}{2}$ inquires, "What angle has a cosine of $\frac{\sqrt{3}}{2}$?". The answers to these questions are angles, typically expressed in radians, that fall within specific ranges defined by the inverse trigonometric functions.
Evaluating sin⁻¹ 0
Let's first address the inverse sine function, denoted as $\\sin^-1} 0$. This function seeks the angle whose sine is 0. Recall that the sine function corresponds to the y-coordinate on the unit circle. The y-coordinate is 0 at two points on the unit circle x$, is restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$ or [-90°, 90°]. Therefore, we must choose the angle within this range. The angle that satisfies both the condition (sine of 0) and the range restriction is 0 radians. Thus, we conclude that $\\sin^{-1} 0 = 0$. This understanding is crucial as it simplifies our original expression, paving the way for further evaluation.
Evaluating cos⁻¹(√3/2)
Next, we turn our attention to the inverse cosine function, $\cos^{-1} \frac{\sqrt{3}}{2}$. This function asks us to find the angle whose cosine is $\frac{\sqrt{3}}{2}$. Recall that the cosine function corresponds to the x-coordinate on the unit circle. The range of the inverse cosine function, $\cos^{-1} x$, is defined as $[0, \pi]$ or [0°, 180°]. Within this range, we look for the angle whose cosine value is $\frac{\sqrt{3}}{2}$. From our knowledge of special trigonometric values, we know that the angle that satisfies this condition is $\\frac{\pi}{6}$ radians (30 degrees). Therefore, we can confidently state that $\cos^{-1} \frac{\sqrt{3}}{2} = \frac{\pi}{6}$. This evaluation is another significant step in simplifying our original expression and bringing us closer to the final answer.
Substituting the Values
Now that we have determined the values of the inverse trigonometric functions, we can substitute them back into the original expression. This substitution is a crucial step in simplifying the expression and making it more manageable. Recall that our original expression is $\sin \left(\sin ^-1} 0-\cos ^{-1} \frac{\sqrt{3}}{2}\right)$. We found that $\sin^{-1} 0 = 0$ and $\cos^{-1} \frac{\sqrt{3}}{2} = \frac{\pi}{6}$. Substituting these values into the expression, we get{6}\right)$. This simplifies to $\sin \left(-\frac{\pi}{6}\right)$, which is a much simpler form to evaluate.
Evaluating sin(-π/6)
To evaluate $\sin \left(-\frac\pi}{6}\right)$, we need to recall the properties of the sine function, particularly its behavior with negative angles. The sine function is an odd function, which means that $\sin(-x) = -\sin(x)$. Applying this property to our expression, we have{6}\right) = -\sin \left(\frac{\pi}{6}\right)$. Now, we need to determine the value of $\sin \left(\frac{\pi}{6}\right)$. From our knowledge of special trigonometric values, we know that $\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$. Therefore, $\sin \left(-\frac{\pi}{6}\right) = -\sin \left(\frac{\pi}{6}\right) = -\frac{1}{2}$. This final step gives us the exact value of the original expression.
Conclusion
In conclusion, by systematically breaking down the expression $\sin \left(\sin ^{-1} 0-\cos ^{-1} \frac{\sqrt{3}}{2}\right)$, we have successfully determined its exact value. We began by evaluating the inverse trigonometric functions individually, finding that $\sin^{-1} 0 = 0$ and $\cos^{-1} \frac{\sqrt{3}}{2} = \frac{\pi}{6}$. Substituting these values into the original expression, we simplified it to $\sin \left(-\frac{\pi}{6}\right)$. Finally, using the property of the sine function for negative angles and our knowledge of special trigonometric values, we found that $\sin \left(-\frac{\pi}{6}\right) = -\frac{1}{2}$. Therefore, the exact value of the expression $\sin \left(\sin ^{-1} 0-\cos ^{-1} \frac{\sqrt{3}}{2}\right)$ is -1/2. This exercise highlights the importance of understanding inverse trigonometric functions, their properties, and trigonometric identities in evaluating complex expressions.
Keywords: Trigonometric expressions, inverse trigonometric functions, sine, cosine, unit circle, radians, special trigonometric values, trigonometric identities, evaluation, exact value.
