Find all values of theta where 0degrees<theta<360 degrees when csc theta = square root 2
Please detail how/why i can get to the solution. I need to understand the concept clearly...... It will probably relate to something taught down the line I'm sure.
see prior post. and geez, wait a bit willya?
given : cscØ = √2
or by definition: sinØ = 1/√2
recall the CAST rule, which says that the sine is positive in quadrants I or II
method 1: recognize the main trig ratios of 0°,30°, 60°, 45°, 90°
we know that sin 45° = 1/√2
so Ø is 45 in quadrant I
or
Ø is 180-45 or 35° in quadrant II
method 2: (if you don't know the main rations)
have your calculator find 1/√2 = .7071...
make sure it is set to degrees,
press 2ndF sin, then =
to get 45
If you are doing a lot of these angle related problems, I suggest you copy in your notebook pictures of the 30-60-90 and the 45-45-90 degree triangles with the ratio of their sides:
1 : √3 : 2 for the first and 1 : 1 : √2 for the second
(notice how I recognized the 1/√2 from the last part ? )
Steve , did not realize you had already given a detailed explanation.
Patience would have indeed been a virtue that could have been exercised by the poster.
Thank you to both of you, I am new to this site and it has been a blessing. Reiny your CAST rule which I was not taught, is very helpful. Your whole post will be useful come my exam. My book is not as straight forward..I fear for my GPA. :(
To find all the values of theta where 0° < theta < 360° and csc theta = sqrt(2), we first need to understand the concept of the cosecant function.
The cosecant function (csc) is the reciprocal of the sine function. It represents the ratio between the hypotenuse and the side opposite to an angle in a right triangle. The formula for the cosecant is csc(theta) = 1 / sin(theta).
Now, let's solve the equation csc theta = sqrt(2):
1) Start by taking the reciprocal of both sides of the equation:
1 / (csc theta) = 1 / (sqrt(2))
2) Invert the right side using the square root property:
1 / (csc theta) = sqrt(2) / 1
1 / (csc theta) = sqrt(2)
3) Take the reciprocal again, getting rid of the denominator:
csc theta = 1 / (sqrt(2))
4) We know that csc(theta) = 1 / sin(theta), so we can rewrite the equation as:
1 / sin(theta) = 1 / (sqrt(2))
5) Since the denominators are the same, we can equate the numerators:
1 = sin(theta) / (sqrt(2))
6) Multiply both sides of the equation by sqrt(2) to get rid of the fraction:
sqrt(2) = sin(theta)
Now, we have found that sin(theta) = sqrt(2). To find all possible values of theta, we can use the unit circle or a calculator to determine the angles where sin(theta) equals sqrt(2).
Using a calculator, the inverse sine function (sin^(-1)) can be applied to both sides of the equation:
theta = sin^(-1)(sqrt(2))
Calculating sin^(-1)(sqrt(2)) on a calculator yields approximately 70.53°.
Since the sine function is periodic with a period of 360°, we need to find all values of theta within the given range of 0° to 360° that satisfy the equation sin(theta) = sqrt(2).
The first solution, theta = 70.53°, is within the given range. To find additional solutions, we can add or subtract a full period of 360° to the initial solution:
theta = 70.53° + 360° = 430.53° (not within the given range)
theta = 70.53° - 360° = -289.47° (not within the given range)
Therefore, the only solution for 0° < theta < 360° where csc theta = sqrt(2) is theta = 70.53°.