given x = 1,110 degrees
Find sinx, cosx, tanx, cotx, secx, cscx, without using a calculator. Each result should be a fraction. No decimals involved...
I don't even know where to begin, i missed a class, so please explain the work fully, Thanks kenny
You have some identity formulas...Look them up in your text...they go something like this:
sinTheta=sin(theta-n*360) and you could use that to find sin1110=sin30
There are others similar.
To find the trigonometric values of x = 1,110 degrees without using a calculator, we'll need to use some trigonometric identities and find reference angles.
First, let's determine the reference angle for 1,110 degrees. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis.
To find the reference angle, we can subtract multiples of 360 degrees until we get a result between 0 and 360 degrees:
1,110 degrees - 360 degrees = 750 degrees (still greater than 360)
750 degrees - 360 degrees = 390 degrees (still greater than 360)
390 degrees - 360 degrees = 30 degrees (within the range of 0 to 360)
So, the reference angle for 1,110 degrees is 30 degrees.
Now, let's find the trigonometric values using the reference angle:
1. sin(x):
Since sin(x) is positive in the first and second quadrants, and the reference angle of 30 degrees is in the first quadrant, we can directly find the value of sin(30) using the unit circle.
The value of sin(30) is 1/2.
Therefore, sin(1,110 degrees) = sin(30 degrees) = 1/2.
2. cos(x):
Similar to sin(x), cos(x) is positive in the first and fourth quadrants. Since the reference angle of 30 degrees is in the first quadrant, we can find the value of cos(30) using the unit circle.
The value of cos(30) is √3/2.
Therefore, cos(1,110 degrees) = cos(30 degrees) = √3/2.
3. tan(x):
To find tan(x), we can use the ratio tan(x) = sin(x) / cos(x).
tan(1,110 degrees) = sin(1,110 degrees) / cos(1,110 degrees)
Using the values we have found, tan(1,110 degrees) = (1/2) / (√3/2) = 1/√3.
4. cot(x):
To find cot(x), we can use the reciprocal of tan(x), which is cot(x) = 1 / tan(x).
cot(1,110 degrees) = 1 / tan(1,110 degrees)
Using the value of tan(1,110 degrees) we found earlier, cot(1,110 degrees) = 1 / (1/√3) = √3.
5. sec(x):
Similar to cos(x), sec(x) is the reciprocal of cos(x). So, sec(x) = 1 / cos(x).
sec(1,110 degrees) = 1 / cos(1,110 degrees)
Using the value of cos(1,110 degrees) we found earlier, sec(1,110 degrees) = 1 / (√3/2) = 2/√3.
6. csc(x):
Similar to sin(x), csc(x) is the reciprocal of sin(x). So, csc(x) = 1 / sin(x).
csc(1,110 degrees) = 1 / sin(1,110 degrees)
Using the value of sin(1,110 degrees) we found earlier, csc(1,110 degrees) = 1 / (1/2) = 2.
Therefore, the trigonometric values for x = 1,110 degrees are:
sin(x) = 1/2
cos(x) = √3/2
tan(x) = 1/√3
cot(x) = √3
sec(x) = 2/√3
csc(x) = 2.