The value of sin78° + sin6° - sin66° - sin42° is?

Steps to Solve

1. Rewrite the Expression

We begin with the expression

$$ \sin 78^\circ + \sin 6^\circ - \sin 66^\circ - \sin 42^\circ $$

2. Use the Co-function Identities

Use the identity $\sin(90^\circ - x) = \cos(x)$.

• For $\sin 78^\circ$, we have:

$$ \sin 78^\circ = \cos 12^\circ $$

• For $\sin 66^\circ$, we see:

$$ \sin 66^\circ = \cos 24^\circ $$

• For $\sin 6^\circ$ and $\sin 42^\circ$, we can leave them as is for now.

3. Substitute Values

Now substitute these values back into the expression:

$$ \cos 12^\circ + \sin 6^\circ - \cos 24^\circ - \sin 42^\circ $$

4. Use Known Values for Sine and Cosine

Recall that:

$$ \sin 6^\circ = \sin(90^\circ - 84^\circ) = \cos 84^\circ $$

Thus the expression becomes:

$$ \cos 12^\circ + \cos 84^\circ - \cos 24^\circ - \sin 42^\circ $$

5. Group Terms

Grouping similarities can help simplify:

$$ (\cos 12^\circ - \cos 24^\circ) + (\cos 84^\circ - \sin 42^\circ) $$

6. Evaluate Values or Use Calculator

Using values for $\sin$ and $\cos$ functions, if needed:

• $\sin 6^\circ \approx 0.1045$
• $\sin 42^\circ \approx 0.6691$
• $\cos 12^\circ \approx 0.9781$
• $\cos 24^\circ \approx 0.9063$
• $\cos 84^\circ \approx 0.1045$

Now we substitute these:

$$ (0.9781 - 0.9063) + (0.1045 - 0.6691) $$

7. Final Calculation

Doing the arithmetic:

1. First part:

$$ 0.9781 - 0.9063 \approx 0.0718 $$

2. Second part:

$$ 0.1045 - 0.6691 \approx -0.5646 $$

Combine both:

$$ 0.0718 - 0.5646 \approx -0.4928 $$

The resulting value is approximately:

$$ -0.5 = - \frac{1}{2} $$