Rewrite the complex number 8 ( cos 2.5 + i sin 2.5 ) in a + b i form

Question

My struggle has been that I cannot understand how to find the exact value of cos and sin and the decimal I get from my calculator doesn't register as correct when I punch it in. If you could also explain how to find the exact value for degrees for sin, cos, and tan that don't fall on the radian graph that would be much appreciated.

Answer 1

The problem is likely that you are using degrees instead of radians. If you don't see the degree symbol, then you should assume the angle measurement is in radians.

The exact value of an angle of EITHER 2.5° or 2.5 radians would be tough to get so just use the calculator value. cos(2.5) = -0.801 (approximately) and sin(2,5) = 0.598 so the a + bi form is 8•(-0.801) + 8•(0.598)i = -6.409 + 4.788i

Answer 2

First of all, the complex number you have submitted is ALREADY in rectangular (a+bi) form:

a= 8 cos 2.5 and b=8 sin 2.5. The polar form would be 8e^2.5i.

The conversion from radians to degrees is r/π = d/180 where r is the radian measure and d is degree measure. For example: 30º = π/6

Your calculator probably has a setting which specifies whether degrees or radians is used and the setting can be changed back and forth.

Remember that the trig functions, in general, are irrational numbers so that you should not expect "exact" values for most of what you do with them.