To find the exact value of cos 10°, we can use the formula for the cosine of a double angle or half angle, but it doesn’t yield a straightforward exact value. However, we can estimate the value using known angles.
Another approach involves using the identity for the cosine of a sum or difference of angles. For instance, we can express 10° as the average of two angles whose exact cosine values we know. One helpful identity is:
cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)
By letting a be 30° and b be 20° or other combinations that aren’t too far off from 10°, we can keep approximating until we get close enough for practical purposes. However, it’s key to note that:
There is no simple rational expression for cos 10° in basic trigonometric terms that relies purely on square roots like those for 30°, 45°, or 60°.
In conclusion, while you cannot directly calculate cos 10° like other angles using elementary trigonometric identities, it’s often approximated in higher mathematics or through series expansions for practical applications. The exact value remains a non-rational number.