Simplify cot(arcsin(-( square root of 5)/6))
The question is asking you to perform a trigonometric simplification involving the composite function of the cotangent (cot) of the inverse sine function (arcsin). Specifically, it requires you to simplify the expression cot(arcsin(-√5/6)), which entails finding the value of the cotangent when the angle is given in the form of an arcsine of a negative fraction, in this case -√5/6. This typically requires understanding reciprocal trigonometric identities, properties of inverse trigonometric functions, and possibly using Pythagorean identities to assist with simplifying the expression to a basic trigonometric value or a simplified radical expression.
$cot \left(\right. arcsin \left(\right. - \frac{\sqrt{5}}{6} \left.\right) \left.\right)$
Answer
Solution:
Step 1:
Apply the product rule for radicals to combine the terms. $- \frac{\sqrt{31 \cdot 5}}{5}$
Step 2:
Perform the multiplication of the numbers under the radical. $- \frac{\sqrt{155}}{5}$
Step 3:
Present the final result in its various forms.
- Exact Form: $- \frac{\sqrt{155}}{5}$
- Decimal Form: $- 2.48997991 \ldots$
Knowledge Notes:
The problem requires simplifying the cotangent of an arcsine function, where the input to the arcsine is a negative fraction involving a square root. The solution provided, however, does not directly address the trigonometric simplification but rather seems to be a simplification of a radical expression. This solution appears to be incorrect or unrelated to the given problem.
To correctly solve the original problem, we would need to use trigonometric identities and properties of inverse trigonometric functions. Here are the relevant knowledge points:
Arcsine Function: The arcsine function, denoted as $\arcsin(x)$, is the inverse of the sine function. It returns the angle whose sine is $x$. The range of arcsine is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
Cotangent Function: The cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side. It is the reciprocal of the tangent function, $\cot(\theta) = \frac{1}{\tan(\theta)}$.
Trigonometric Identities: To simplify expressions involving trigonometric functions, we often use identities such as $\sin^2(\theta) + \cos^2(\theta) = 1$ and $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.
Simplifying Trigonometric Expressions: When simplifying an expression like $\cot(\arcsin(x))$, we would typically express the cotangent in terms of sine and cosine, and then use the Pythagorean identity to find the missing side of a right triangle represented by the arcsine function.
Inverse Trigonometric Functions: The inverse trigonometric functions can be represented in terms of right triangles, where the output of the function is an angle, and the input is a ratio of sides.
Given the discrepancy between the problem statement and the provided solution, it seems necessary to re-evaluate the problem-solving process to address the trigonometric nature of the problem. The correct approach would involve defining a right triangle based on the arcsine value given and then using trigonometric identities to find the cotangent of the resulting angle.
