To explore whether the product of sin 30 degrees and sin 60 degrees is the same as the product of cos 60 degrees tan 30 degrees and cos 30 degrees tan 60 degrees, we first need to calculate each side of the equation.
Calculating sin 30 and sin 60
We know that:
- sin 30 degrees = 1/2
- sin 60 degrees = √3/2
Thus, the product of sin 30 degrees and sin 60 degrees can be calculated as:
sin 30 degrees * sin 60 degrees = (1/2) * (√3/2) = √3/4
Calculating cos 60, tan 30, cos 30, and tan 60
Next, let’s calculate the values for cos 60 degrees, tan 30 degrees, cos 30 degrees, and tan 60 degrees:
- cos 60 degrees = 1/2
- tan 30 degrees = 1/√3
- cos 30 degrees = √3/2
- tan 60 degrees = √3
Now we can compute the product:
(cos 60 * tan 30) * (cos 30 * tan 60)
Calculating each part, we have:
cos 60 * tan 30 = (1/2) * (1/√3) = 1/(2√3)
cos 30 * tan 60 = (√3/2) * (√3) = 3/2
Therefore, the total product becomes:
(1/(2√3)) * (3/2) = 3/(4√3)
Comparing the Results
Now, looking back at the two products, we have:
- sin 30 * sin 60 = √3/4
- cos 60 tan 30 * cos 30 tan 60 = 3/(4√3)
Finding a Common Denominator
To compare these two fractions, let’s convert 3/(4√3) to have a common denominator with √3/4:
√3/4 = √3 * √3/(4 * √3) = 3/(4√3)
This shows that:
Conclusion
√3/4 = 3/(4√3), thus verifying that:
sin 30 degrees * sin 60 degrees = cos 60 degrees tan 30 degrees * cos 30 degrees tan 60 degrees