Let's break down the solution step by step:
### Assertion (A):
We need to find the value of the expression:
[tex]\[ \sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ \][/tex]
Step 1: Calculate [tex]\(\sin 60^\circ\)[/tex] and [tex]\(\cos 60^\circ\)[/tex]:
- [tex]\(\sin 60^\circ = 0.8660254037844386\)[/tex]
- [tex]\(\cos 60^\circ = 0.5000000000000001\)[/tex]
Step 2: Calculate [tex]\(\sin 30^\circ\)[/tex] and [tex]\(\cos 30^\circ\)[/tex]:
- [tex]\(\sin 30^\circ = 0.49999999999999994\)[/tex]
- [tex]\(\cos 30^\circ = 0.8660254037844387\)[/tex]
### Substituting these values into the expression:
[tex]\[ \sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ \][/tex]
[tex]\[ = (0.8660254037844386) \times (0.8660254037844387) + (0.49999999999999994) \times (0.5000000000000001) \][/tex]
### Performing the calculations:
[tex]\[ = 0.7500000000000001 + 0.25 \][/tex]
[tex]\[ = 1.0 \][/tex]
Therefore, the value of the expression [tex]\(\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ\)[/tex] is indeed [tex]\(1.0\)[/tex].
### Reason (R):
The values given in the reason are:
- [tex]\(\sin 90^\circ = 1.0\)[/tex]
- [tex]\(\cos 90^\circ = 0.0\)[/tex]
These values are fundamental trigonometric identities and true by definition:
- The sine of 90 degrees is always 1.
- The cosine of 90 degrees is always 0.
### Conclusion:
The assertion (A) that the value of [tex]\(\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ\)[/tex] is 1 is correct. The reason (R) gives correct trigonometric values but does not explain the assertion directly – the main connection is through the use of trigonometric identities involving 90 degrees.
Based on the calculations and the given trigonometric facts, both the assertion (A) and the reason (R) are true, but the reason does not directly explain the assertion.
