Understanding the Trigonometric Product Question

The question asks for the value of the product of cosine functions for a sequence of angles starting from 46 degrees and increasing by 1 degree, indicated by the ellipsis (…).

The expression is: \(\cos 46^\circ \cos 47^\circ \cos 48^\circ \cos 49^\circ \cos 50^\circ \dots\)

This means we are multiplying the cosine of 46 degrees, the cosine of 47 degrees, the cosine of 48 degrees, and so on, for consecutive integer degree values.

Analyzing the Sequence of Angles

The angles in the product form a sequence: \(46^\circ, 47^\circ, 48^\circ, 49^\circ, 50^\circ, 51^\circ, \dots\)

Since the angles are increasing by 1 degree in each step, this sequence of angles will eventually include \(90^\circ\). Let's look at the sequence further:

\(46^\circ\)

\(47^\circ\)

\(48^\circ\)

...

\(89^\circ\)

\(90^\circ\)

\(91^\circ\)

...

The product includes the cosine of each of these angles.

Identifying a Key Term in the Product

As we saw, the sequence of angles \(46^\circ, 47^\circ, 48^\circ, \dots\) includes the angle \(90^\circ\). Therefore, one of the terms in the product \(\cos 46^\circ \cos 47^\circ \cos 48^\circ \dots\) will be \(\cos 90^\circ\).

Evaluating the Key Term: Cosine of 90 Degrees

We need to recall the standard value of the cosine function for key angles in trigonometry. The value of \(\cos 90^\circ\) is a well-known trigonometric value.

The value of \(\cos 90^\circ\) is \(0\).

This can be understood from the unit circle (the x-coordinate of the point at \(90^\circ\)) or the graph of the cosine function.

So, we have a term equal to \(0\) in our product.

Calculating the Value of the Product

The product is \(\cos 46^\circ \times \cos 47^\circ \times \cos 48^\circ \times \dots\). Since the angle \(90^\circ\) is included in the sequence of angles, the term \(\cos 90^\circ\) is part of this multiplication.

The product can be written as:

\(\cos 46^\circ \times \cos 47^\circ \times \dots \times \cos 89^\circ \times \cos 90^\circ \times \cos 91^\circ \times \dots\)

Substituting the value of \(\cos 90^\circ = 0\):

\(\cos 46^\circ \times \cos 47^\circ \times \dots \times \cos 89^\circ \times 0 \times \cos 91^\circ \times \dots\)

When any number, or a product of numbers, is multiplied by zero, the result is always zero.

Therefore, the value of the entire product \(\cos 46^\circ \cos 47^\circ \cos 48^\circ \cos 49^\circ \cos 50^\circ \dots\) is \(0\).

Conclusion

The product \(\cos 46^\circ \cos 47^\circ \cos 48^\circ \dots\) includes the term \(\cos 90^\circ\). Since \(\cos 90^\circ = 0\), the value of the entire product is \(0\). This matches option 2.

Revision Table: Key Trigonometric Values

Additional Information: Product of Cosines

Problems involving the product of trigonometric functions over a sequence of angles often rely on identifying specific angles where the function takes a value of 0 or 1, or using trigonometric identities.

In this specific case, the inclusion of \(\cos 90^\circ\) simplifies the problem significantly because \(\cos 90^\circ = 0\). Any product containing a factor of 0 will result in 0.

Other common angles where cosine is zero include \(270^\circ\), \(450^\circ\), etc. (i.e., \(90^\circ + n \times 180^\circ\) for integer \(n\)). If the sequence of angles included any of these, the product would also be zero.

For products of sines, if the sequence of angles included \(0^\circ\) or \(180^\circ\) (or multiples of \(180^\circ\)), the product would be zero because \(\sin 0^\circ = 0\) and \(\sin 180^\circ = 0\).