To find the exact value of the given expressions, we will need to evaluate each trigonometric function separately.
a) csc(1/√2)
The cosecant function is the reciprocal of the sine function, so:
csc(x) = 1/sin(x)
Now, if we take x = 1/√2, we look for sin(1/√2). This angle does not correspond to a standard angle on the unit circle, so we will evaluate sin(1/√2) numerically. However, it’s essential to interpret the input correctly because the value appears misrepresented as an angle. If 1/√2 is treated strictly as a number rather than an angle, we cannot find an exact value for csc(1/√2). Generally, csc(x) is calculated for angles, not for irrational numbers. Therefore, this value is not standard.
b) cos(1/√3 * 2)
Similar to csc, the cosine function is also evaluated for angles. In cos(1/√3 * 2), we first need to calculate 1/√3 * 2. This equals approximately 1.1547, which again is not a standard angle. Thus, we would need to evaluate cos(1.1547). For practical purposes, we can either compute this numerically or use a calculator. The exact mathematical expression itself, however, does not yield a straightforward answer in standard trigonometric ratios.
In conclusion, such expressions should typically be addressed with specific angles or exact known values for precise calculations. Further clarification on how angles are intended to be interpreted would be useful for providing a more accurate evaluation.