Write the expression as the sine or cosine of an angle.

[tex]\[
\sin 42^{\circ} \cos 17^{\circ} - \cos 42^{\circ} \sin 17^{\circ}
\][/tex]

Hint:

[tex]\[
\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B
\][/tex]

[tex]\[
\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B
\][/tex]



Answer :

To express the given trigonometric expression as a sine or cosine of an angle, we can use one of the angle subtraction identities. The hint provided suggests using the sine angle subtraction formula:

[tex]\[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \][/tex]

Given the expression:

[tex]\[ \sin 42^{\circ} \cos 17^{\circ} - \cos 42^{\circ} \sin 17^{\circ} \][/tex]

We can identify it as fitting the form of the sine angle subtraction identity:

[tex]\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \][/tex]

By comparing terms, we see that:
- [tex]\( A = 42^{\circ} \)[/tex]
- [tex]\( B = 17^{\circ} \)[/tex]

Therefore, we substitute [tex]\( A \)[/tex] and [tex]\( B \)[/tex] into the identity:

[tex]\[ \sin 42^{\circ} \cos 17^{\circ} - \cos 42^{\circ} \sin 17^{\circ} = \sin(42^{\circ} - 17^{\circ}) \][/tex]

Perform the subtraction inside the sine function:

[tex]\[ 42^{\circ} - 17^{\circ} = 25^{\circ} \][/tex]

Thus, the given expression can be written as:

[tex]\[ \sin 42^{\circ} \cos 17^{\circ} - \cos 42^{\circ} \sin 17^{\circ} = \sin 25^{\circ} \][/tex]

So the expression is:

[tex]\[ \sin 25^{\circ} \][/tex]