Trigonometric Substitution
a. Explain why the substitution x = asin(Theta) guarantees that cos(Theta) is greater than or equal to 0 .
b. What sort of restriction must we put on x and a so that the substitution x = asec(Theta)will guarantee that tan(Theta) is greater than or equal to 0?
x=1/2tan0 write f(X)= x/1+4
a. To explain why the substitution x = asin(θ) guarantees that cos(θ) is greater than or equal to 0, we need to consider the properties of the trigonometric functions and their ranges.
In the unit circle, the sine function is defined as the y-coordinate of the point on the unit circle corresponding to a given angle, θ. The range of the sine function is from -1 to 1. This means that asin(θ) will yield values of x between -a and a.
The cosine function, on the other hand, is defined as the x-coordinate of the point on the unit circle corresponding to a given angle, θ. The range of the cosine function is also from -1 to 1. However, in the first and second quadrants of the unit circle, the cosine function is positive (greater than 0).
By substituting x = asin(θ), we are effectively restricting the values of x to lie between -a and a. Since asin(θ) is mainly defined in the first and fourth quadrants of the unit circle, the corresponding values of θ will be in such a way that cos(θ) is greater than or equal to 0. This is because in those quadrants, cos(θ) is positive.
Therefore, the substitution x = asin(θ) guarantees that cos(θ) is greater than or equal to 0.
b. To guarantee that tan(θ) is greater than or equal to 0 with the substitution x = asec(θ), we need to consider the behavior of tangent and secant functions.
The tangent function is defined as the ratio of the sine and cosine functions, tan(θ) = sin(θ) / cos(θ). It is positive when both the sine and cosine functions have the same sign (either both positive or negative). In the first and third quadrants of the unit circle, tangent is positive.
The secant function is defined as the reciprocal of the cosine function, sec(θ) = 1 / cos(θ). Its range is from -∞ to -1 and from 1 to ∞, excluding the values between -1 and 1.
By substituting x = asec(θ), the resulting values for x will have a range that includes positive numbers. However, since sec(θ) is not defined for values between -1 and 1, we need to ensure that x does not take values within that range. Therefore, we must impose the restriction that a is greater than or equal to 1 to avoid those values of x.
With this restriction, the substitution x = asec(θ) ensures that tan(θ) is greater than or equal to 0 because sec(θ) is positive in the first and second quadrants, and tangent is positive in the first and third quadrants of the unit circle.