Write the given expression as the cosine of an angle. cos 60° cos 65° − sin 60° sin 65°
you should recognize this as the use of
cos(A+B)
cos 60° cos 65° − sin 60° sin 65°
= cos(60°+65°)
= cos 125°
Well, if you're looking for the cosine of an angle for this expression, we can use a little trigonometric creativity. Let me put on my clown wig and get to work!
We can use the cosine of the difference formula: cos(A - B) = cos A cos B + sin A sin B. Now, let's identify our A and B. We have cos 60° cos 65° - sin 60° sin 65°.
For the first term, let's let A = 60° and B = 65°. That gives us cos(60° - 65°), which is equal to cos(-5°). But wait, the cosine function is even, meaning that cos(-x) = cos(x). So, we have cos(5°) for the first term.
Now for the second term, let's interchange the values of A and B. So, A = 65° and B = 60°. That gives us cos(65° - 60°). Simplifying, we have cos(5°) again.
Putting it all together, we have cos(5°) - cos(5°). And you know what? When you subtract something from itself, you get zero! So, the given expression can be written as...drumroll, please...0!
And just like that, we've turned a trigonometric expression into a hilarious math joke. Thanks for the fun question!
To express the given expression as the cosine of an angle, we can use the cosine of a difference formula.
The formula is: cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
In this case, A = 60° and B = 65°.
Substituting the values into the formula, we have:
cos(60° - 65°) = cos(60°)cos(65°) + sin(60°)sin(65°)
Simplifying further:
cos(-5°) = cos(60°)cos(65°) + sin(60°)sin(65°)
Since the cosine function is an even function, it follows that cos(-5°) = cos(5°).
Therefore, the given expression can be written as:
cos(5°) = cos(60°)cos(65°) + sin(60°)sin(65°)
To express the given expression as the cosine of an angle, we can use the trigonometric identity for the cosine of the difference of two angles.
The trigonometric identity for the cosine of the difference of two angles states:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
In the given expression, we have:
cos 60° cos 65° − sin 60° sin 65°
Using the trigonometric identity, we can rewrite it as:
cos(60° - 65°)
Simplifying, we have:
cos(-5°)
Therefore, the given expression is equivalent to cos(-5°), or simply the cosine of -5 degrees.