How do you remove the 2 pi in this problem?
csc 19pi/4 remove 2 pi -> (8pi/4) -> 11pi/4 -> 3pi/4
>> i really don't know how do they the 2 pi twice so it would become 8pi.
Please help me!
> i really don't know how do they REMOVE the 2 pi twice so it would become 8pi.
Please help me!
>>sorry, please help me
19π/4
= 4π + 3π/4
then
csc (19π/4)
= csc (4π + 3π/3)
since csc 19π/4
= cos 19π/4 + i sin 19π/4
= cos (4π + 3π/4) + i sin (4π + 3π/4)
since both the cosine and the sine function have periods of 2π , adding or subtracting multiples of 2π will give us the same answer
so let's "remove " 4π
= cos 3π/4 + i sin 3π/4
= csc (3π/4)
check:
csc 19π/4 = -1/√2 + i 1/√2
csc 3π/4 = -1/√2 + i 1/√2
I really have no idea what
csc 19pi/4 remove 2 pi -> (8pi/4) -> 11pi/4 -> 3pi/4
is supposed to mean.
unless the did this:
19π/4 - 2π = 11π/4
11π/4 - 2π = 3π/4 , which is what I did above.
the 8π/4 does not belong, it could just be a typo
To understand how the 2 pi is removed twice in the given problem, let's break it down step by step.
The given expression is:
csc(19π/4)
To simplify this, we need to recall the unit circle and identify the angle 19π/4.
In the unit circle, every 2π (or one full revolution) corresponds to completing one cycle of the trigonometric function. So, we can remove any number of complete cycles without changing the value of the trigonometric function.
In this case, the angle 19π/4 is equal to 4 complete cycles plus an additional 3π/4.
Now, consider the expression:
csc(19π/4)
Since we have 4 complete cycles (8π) plus an additional 3π/4, we can rewrite this as:
csc(8π + 3π/4)
By using the periodicity of the trigonometric functions, we know that the value of the trigonometric function at an angle plus a multiple of 2π is the same as the value at that angle. Therefore, we can rewrite the expression again:
csc(3π/4)
Now, we have reduced the problem to finding the csc of 3π/4, which is an angle within one cycle (0 to 2π). This angle is within the first quadrant, and we can find the sine function (sin) at this angle to be positive.
Using the unit circle, we find that sin(3π/4) is equal to √2/2.
Finally, to find the csc of 3π/4, we take the reciprocal of sin(3π/4):
csc(3π/4) = 1/sin(3π/4) = 1/(√2/2) = 2/√2 = √2.
So, the final answer is √2.
To summarize the steps:
1. Identify the given angle on the unit circle (19π/4).
2. Recognize that 19π/4 is equivalent to 4 complete cycles (8π) plus an additional angle (3π/4).
3. Rewrite the expression as csc(8π + 3π/4).
4. Use the periodicity of the trigonometric functions to simplify the expression to csc(3π/4).
5. Determine the sine of 3π/4 from the unit circle (√2/2).
6. Take the reciprocal of sin(3π/4) to find the csc(3π/4) (√2).
7. The final answer is √2.