for what values of theta are the tangent 0?for what values of theta is the tangent undefined?explain.
The tangent function is defined as the ratio of the sine of an angle to its cosine. To find the values of theta where the tangent function is zero, we need to look for the angles where the sine of that angle is zero.
The sine function is zero at specific angles, which occur at multiples of pi. So, the values of theta where the tangent is zero are given by theta = n * pi, where n is an integer.
On the other hand, the tangent function is undefined when the cosine of an angle is zero. The cosine function is zero at multiples of pi/2. Therefore, the values of theta where the tangent is undefined are given by theta = (n + 1/2) * pi, where n is an integer.
To summarize:
- The tangent is zero at theta = n * pi, where n is an integer.
- The tangent is undefined at theta = (n + 1/2) * pi, where n is an integer.
To find the values of theta where the tangent function is equal to zero and where it is undefined, we need to consider the properties of the tangent function.
1. Values of theta where the tangent is zero:
The tangent function is equal to zero when the numerator sin(theta) is zero. Sin(theta) equals zero at theta = 0, pi, 2pi, 3pi, and so on. This is because at these angles, the y-coordinate of a point on the unit circle is zero. So, the values of theta where the tangent is zero are given by:
theta = 0 + n*pi, where n is an integer.
2. Values of theta where the tangent is undefined:
The tangent function is undefined when the denominator cos(theta) is zero. Cos(theta) equals zero at theta = pi/2, 3pi/2, 5pi/2, and so on. This is because at these angles, the x-coordinate of a point on the unit circle is zero. The tangent function is defined as the ratio of sin(theta) to cos(theta), and division by zero is undefined. So, the values of theta where the tangent function is undefined are given by:
theta = pi/2 + n*pi, where n is an integer.
In summary:
- The tangent function is equal to zero at theta = 0 + n*pi, where n is an integer.
- The tangent function is undefined at theta = pi/2 + n*pi, where n is an integer.
a. Tan (theta) = 0, Theta = 0.
b. Theta = 90o.