Question

What is the exact value of cos 80 cos 20 + sin 80 sin 20 show all steps

Answer 1

Explanation:

Ok, so first things first, we need to remember that cosine and sine are just different ways of measuring angles. Cosine measures the relationship between the side of a triangle that is opposite the angle and the hypotenuse (the longest side), while sine measures the relationship between the side of a triangle that is opposite the angle and the side that is adjacent (next to) the angle.

Now, let's start with the first part of the equation: cos 80. To find this, we can use a little trick called "angle addition formula" which states that cos (x + y) = cos x cos y - sin x sin y. So, we can write cos 80 as cos (70 + 10) = cos 70 cos 10 - sin 70 sin 10.

Next, we can use the same trick to find cos 20. So, we can write cos 20 as cos (10 + 10) = cos 10 cos 10 - sin 10 sin 10.

Now, we have the first part of the equation: cos 80 cos 20.

Let's move on to the second part: sin 80 sin 20. Just like before, we can use angle addition formula to find sin 80 and sin 20. So, we can write sin 80 as sin (70 + 10) = sin 70 cos 10 + cos 70 sin 10. Similarly, we can write sin 20 as sin (10 + 10) = sin 10 cos 10 + cos 10 sin 10.

Now, we have the second part of the equation: sin 80 sin 20.

Finally, we can add the two parts together:

cos 80 cos 20 + sin 80 sin 20 = (cos 70 cos 10 - sin 70 sin 10) (cos 10 cos 10 - sin 10 sin 10) + (sin 70 cos 10 + cos 70 sin 10) (sin 10 cos 10 + cos 10 sin 10)

So, the exact value of cos 80 cos 20 + sin 80 sin 20 is (cos 70 cos 10 - sin 70 sin 10) (cos 10 cos 10 - sin 10 sin 10) + (sin 70 cos 10 + cos 70 sin 10) (sin 10 cos 10 + cos 10 sin 10)

Keep in mind that the answer depends on the values of the angles and can be calculated by using the values of sine and cosine of the angles.