Find all values of theta where 0degrees<theta<360 degrees when csc theta = square root 2
oh well. Clearly you aren't going to try.
45 and 135
I don't want just the answer but how to get to it. I need to fully understand how to arrive at the conclusion....And that is clearly where I am struggling.
Can you please detail why that is the answer?
review your basic angles. They are
0,30,45,60,90 in QI.
the values for 45 are
sin=1/√2
cos=1/√2
tan = cot = 1
csc=√2
sec=√2
This is because for 45 degrees, the sides of the right triangle in the ratio 1:1:√2
The 30-60-90 triangle has sides 1,√3,2, so you can figure the functions there.
If the angle is drawn in standard position (with the base along the x-axis, with one corner at the origin), then
sin = y/r
cos = x/r
tan = y/x
etc.
r is the hypotenuse, so it is always positive.
Since y is positive in QI and QII, so are sin and csc.
I'm sure this is all explained in greater detail, with examples, in your text.
To find all the values of theta where 0 degrees < theta < 360 degrees when csc(theta) = √2, we can use the reciprocal trigonometric identity.
The reciprocal of csc(theta) is sin(theta), so we have sin(theta) = 1/√2.
To find the values of theta that satisfy this equation, we can use the unit circle or a calculator. In this case, we'll use a calculator.
First, find the inverse sine (or arcsin) of 1/√2. This will give us the principal value of theta.
arcsin(1/√2) ≈ 45 degrees.
This means that one of the solutions is theta = 45 degrees.
Now, since sine is positive in the first and second quadrants, we need to find the other value of theta in the second quadrant.
To find this value, subtract the principal value from 180 degrees:
180 degrees - 45 degrees = 135 degrees.
Therefore, another solution is theta = 135 degrees.
Since we are looking for values of theta between 0 degrees and 360 degrees, these two solutions satisfy the given conditions.
Therefore, the values of theta that satisfy the equation 0 degrees < theta < 360 degrees when csc(theta) = √2 are theta = 45 degrees and theta = 135 degrees.