State the restrictions on the variables for these trigonometric identities.
a)(1 + 2 sin x cos x)/ (sin x + cos x) = sin x + cos x
b) sin x /(1+ cos x) = csc x - cot x
generally, restrictions on variables are caused by denominators being zero
so in #1
sinx + cosx = 0 causes problems
sinx = -cosx
sinx/cosx = -1
tanx = -1 , x is in quadrants II or IV
x = 135 or 225 degrees
or 3pi/4 or 5pi/4 radians
#2.
first of all on the right side we have
cscx and cotx
sinxe cscx = 1/sinx, and cotx = cosx/sinx , whenever the sinx = 0 we have an undefined value
sinx = 0 for x = 0, 180 , and 360 degrees or 0, pi, 2pi radians
secondly we have to worry about 1+cosx
1 + cosx = 0
cosx = -1
x = 180 degrees or pi radians
so the restrictions are 0, pi, 2pi radians, or
0, 180 and 360 degrees
a) To find the restrictions on the variables for the trigonometric identity:
1 + 2 sin x cos x / (sin x + cos x) = sin x + cos x
We need to consider the denominators sin x + cos x and cos x + 1.
1. sin x + cos x:
The sum of sin x and cos x can take any value between -√2 and √2, as sin x and cos x both range from -1 to 1.
Therefore, there are no restrictions on sin x + cos x.
2. cos x + 1:
Since cos x ranges between -1 and 1, cos x + 1 ranges between 0 and 2.
Therefore, the denominator cos x + 1 cannot be zero, so we exclude the values of x where cos x = -1.
Thus, the restrictions on x for this trigonometric identity are:
x ≠ (2n + 1)π, where n is an integer.
b) To find the restrictions on the variables for the trigonometric identity:
sin x / (1 + cos x) = csc x - cot x
We need to consider the denominator 1 + cos x.
1. 1 + cos x:
The value of cos x ranges between -1 and 1.
Therefore, the denominator 1 + cos x ranges between 0 and 2.
Thus, the denominator cannot be zero, so we exclude the values of x where cos x = -1.
Therefore, the restriction on x for this trigonometric identity is:
x ≠ (2n + 1)π, where n is an integer.
To determine the restrictions on the variables in each trigonometric identity, we need to consider the denominators of the fractions involved.
a) (1 + 2 sin x cos x) / (sin x + cos x) = sin x + cos x
For the denominator (sin x + cos x), we know that it cannot equal zero because division by zero is undefined. Therefore, we need to find the values of x that make sin x + cos x = 0.
Rearranging the equation, we have cos x = -sin x. Dividing both sides by cos x, we get tan x = -1.
The values of x that satisfy tan x = -1 are x = (2n + 1) * (pi/4), where n is an integer.
So, the restriction on the variable x for this trigonometric identity is x ≠ (2n + 1) * (pi/4), where n is an integer.
b) sin x / (1 + cos x) = csc x - cot x
For the denominator (1 + cos x), we also need to determine the values of x that make it equal to zero. However, we should also consider the domain of the trigonometric functions csc x and cot x, which have their own restrictions.
The function csc x is equal to 1/sin x. Since the sine function is undefined at multiples of pi, we need to exclude those values from our domain. Therefore, the restriction on x is x ≠ n * pi, where n is an integer.
The function cot x is equal to cos x/sin x. Since the sine function is undefined at multiples of pi, and the cosine function is undefined at odd multiples of pi/2, we need to exclude those values as well. Thus, the restriction on x for cot x is x ≠ (2n + 1) * (pi/2), where n is an integer.
Combining both restrictions, the overall restriction for this trigonometric identity is x ≠ n * pi, (2n + 1) * (pi/2), where n is an integer.
Remember to always check for additional restrictions that may arise from other trigonometric functions in any given equation or identity.