Please Do This And Explain It .
Simplify without using calculator leaving your answer in surd form.
sin 480 - cos 765
........................................
tan 225- cos(- 330)
add/subtract 360 until you get to an angle you recognize
sin480 = sin120 = sin60 = √3/2
cos765 = cos45 = 1/√2
tan225 = 1
cos -330 = cos30 = √3/2
now plug and chug
To simplify the expression without using a calculator and leaving the answer in surd form, we need to remember the trigonometric values of certain angles.
1. Simplify sin 480 - cos 765:
- First, let's convert the given angles to their equivalent angles within the unit circle:
- 480 degrees is equivalent to 480 - 360 = 120 degrees.
- 765 degrees is equivalent to 765 - 720 = 45 degrees.
- Now, we can simplify the expression:
- sin 120 = √3/2
- cos 45 = √2/2
- Therefore, sin 480 - cos 765 = √3/2 - √2/2
2. Simplify tan 225 - cos(-330):
- Again, let's convert the given angles to their equivalent angles within the unit circle:
- 225 degrees lies in the third quadrant, so we add 180 degrees to it, resulting in 225 + 180 = 405 degrees.
- -330 degrees is equivalent to -330 + 360 = 30 degrees.
- Now, we can simplify the expression:
- tan 405 = tan (405 - 360) = tan 45 = 1
- cos 30 = √3/2 (cosine of an angle is positive in the fourth quadrant)
- Therefore, tan 225 - cos(-330) = 1 - √3/2
So the simplified expression is:
sin 480 - cos 765 = √3/2 - √2/2
tan 225 - cos(-330) = 1 - √3/2