Verified answer

Step-by-step explanation:

To simplify \(\sin(61^\circ) - \cos(39^\circ)\), you can use trigonometric identities.

First, let's express both angles in terms of their complements:

\(\sin(61^\circ) = \cos(29^\circ)\) because \(61^\circ + 29^\circ = 90^\circ\).

Now, you can rewrite the expression as:

\(\cos(29^\circ) - \cos(39^\circ)\).

Next, you can use the trigonometric identity \(\cos(A) - \cos(B) = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)\):

\(\cos(29^\circ) - \cos(39^\circ) = -2\sin\left(\frac{29^\circ+39^\circ}{2}\right)\sin\left(\frac{29^\circ-39^\circ}{2}\right)\).

Now, calculate the values inside the sine functions:

\(\sin(34^\circ)\sin(-5^\circ)\).

Since \(\sin(-5^\circ) = -\sin(5^\circ)\), the expression simplifies to:

\(-\sin(34^\circ)\sin(5^\circ)\).

Now, you can calculate the values:

\(-\sin(34^\circ) \cdot \sin(5^\circ) \approx -0.559\).

So, \(\sin(61^\circ) - \cos(39^\circ)\) simplifies to approximately -0.559.

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Sure, here's the full solution for simplifying sin(61°) - cos(39°):

1. First, we'll find the values of sin(61°) and cos(39°):

• sin(61°) ≈ 0.8746
• cos(39°) ≈ 0.7660

2. Now, subtract cos(39°) from sin(61°):

sin(61°) - cos(39°)

≈ 0.8746 - 0.7660

[tex]{ \red{ \boxed{ \tt \: ≈ 0.1086}}}[/tex]

So, sin(61°) - cos(39°) is approximately equal to 0.1086.

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[tex]{ \orange{ \sf \: hope \: it \: helps \: you}}[/tex]