Codculate the value of the fellowing without the use of calculater () $$ \cos 75 $$ क) $$ \sin 70 \sin 10+\operatorname{cociocos} 70 $$

Question

UpStudy AI · Accepted Answer

To calculate $$ \cos 75^\circ $$ using the expression $$ \sin 70^\circ \sin 10^\circ + \cos 70^\circ \cos 10^\circ $$, we can use the cosine addition formula:

$$
\cos(a + b) = \cos a \cos b - \sin a \sin b
$$

However, we can also use the cosine subtraction formula:

$$
\cos(a - b) = \cos a \cos b + \sin a \sin b
$$

In this case, we can rewrite $$ \cos 75^\circ $$ as:

$$
\cos 75^\circ = \cos(70^\circ + 5^\circ)
$$

Using the cosine addition formula:

$$
\cos(70^\circ + 5^\circ) = \cos 70^\circ \cos 5^\circ - \sin 70^\circ \sin 5^\circ
$$

But we can also express $$ \cos 75^\circ $$ in terms of $$ \sin 70^\circ \sin 10^\circ + \cos 70^\circ \cos 10^\circ $$:

$$
\cos 75^\circ = \sin 70^\circ \sin 10^\circ + \cos 70^\circ \cos 10^\circ
$$

Now, we can calculate $$ \sin 70^\circ \sin 10^\circ + \cos 70^\circ \cos 10^\circ $$:

1. **Calculate $$ \sin 70^\circ \sin 10^\circ $$**:
   - We know that $$ \sin 70^\circ = \cos 20^\circ $$ (since $$ \sin(90^\circ - x) = \cos x $$).
   - Therefore, $$ \sin 70^\circ \sin 10^\circ = \cos 20^\circ \sin 10^\circ $$.

2. **Calculate $$ \cos 70^\circ \cos 10^\circ $$**:
   - We know that $$ \cos 70^\circ = \sin 20^\circ $$.
   - Therefore, $$ \cos 70^\circ \cos 10^\circ = \sin 20^\circ \cos 10^\circ $$.

Now we can combine these results:

$$
\cos 75^\circ = \sin 70^\circ \sin 10^\circ + \cos 70^\circ \cos 10^\circ = \cos 20^\circ \sin 10^\circ + \sin 20^\circ \cos 10^\circ
$$

Using the sine addition formula:

$$
\sin(a + b) = \sin a \cos b + \cos a \sin b
$$

We can rewrite it as:

$$
\cos 75^\circ = \sin(20^\circ + 10^\circ) = \sin 30^\circ
$$

Since $$ \sin 30^\circ = \frac{1}{2} $$, we have:

$$
\cos 75^\circ = \frac{1}{2}
$$

Thus, the value of $$ \cos 75^\circ $$ is:

$$
\boxed{\frac{1}{2}}
$$
The value of $$ \cos 75^\circ $$ is $$ \frac{1}{2} $$.

Codculate the value of the fellowing without the use of calculater () \( \cos 75 \) क) \( \sin 70 \sin 10+\operatorname{cociocos} 70 \)

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Trigonometry Questions