Which trigonometric functions are equivalent to tan(theta)? Select all that apply. (2 answers)
a. -tan(-theta)****
b. -tan(theta)
c. tan(-theta)
d. tan(theta+2pi)****
e. tan(theta+pi/2)
A) -tan (-theta)
and
D) tan (theta + 2pi)
Oh, trigonometry, bringing back memories of my wild circus days! So, let's see which of these options can be friends with tan(theta):
a. -tan(-theta) - Well, negative times negative is positive, and tan(theta) is a positive clown, so they could get along! (Correct)
b. -tan(theta) - Ah, this is a negative clown, not quite the same as tan(theta). (Incorrect)
c. tan(-theta) - Oh no, throwing a negative sign at poor tan(theta) might make it go all wobbly! (Incorrect)
d. tan(theta+2pi) - Oh, adding a full circle (2pi) to theta won't change our friendly clown! (Correct)
e. tan(theta+pi/2) - Now, adding half a circle (pi/2) will make our clown spin around and change shape! (Incorrect)
So the answers are a. -tan(-theta) and d. tan(theta+2pi). Keep spreading the trigonometric fun, my friend!
The trigonometric functions that are equivalent to tan(theta) are:
a. -tan(-theta)
d. tan(theta+2pi)
To determine which trigonometric functions are equivalent to tan(theta), we can examine the properties and definitions of the trigonometric functions.
The tangent function (tan) is defined as the ratio of the sine function (sin) to the cosine function (cos). Therefore, we can use this definition to find the equivalent trigonometric functions.
a. -tan(-theta): This is equivalent to -((-sin(-theta))/(cos(-theta))). By taking into account the properties of sin(-theta) and cos(-theta), we can rewrite this as (sin(theta))/(cos(theta)). Since this matches the definition of tan(theta), this is an equivalent function. Therefore, a is correct.
b. -tan(theta): This is simply -tan(theta). Since it is not an equivalent function, b is incorrect.
c. tan(-theta): This is equivalent to (sin(-theta))/(cos(-theta)). By taking into account the properties of sin(-theta) and cos(-theta), we can rewrite this as -(sin(theta))/(cos(theta)). Since this is the negative of tan(theta), it is not an equivalent function. Therefore, c is incorrect.
d. tan(theta+2pi): This is equivalent to (sin(theta+2pi))/(cos(theta+2pi)). By using the periodicity properties of sin and cos (2pi is one period), we can rewrite this as (sin(theta))/(cos(theta)). Since this matches the definition of tan(theta), this is an equivalent function. Therefore, d is correct.
e. tan(theta+pi/2): This is equivalent to (sin(theta+pi/2))/(cos(theta+pi/2)). By using the periodicity properties of sin and cos (pi/2 is one quarter of a period), we can rewrite this as (cos(theta))/(sin(theta)). This is the reciprocal of tan(theta), so it is not an equivalent function. Therefore, e is incorrect.
Therefore, the trigonometric functions equivalent to tan(theta) are a. -tan(-theta) and d. tan(theta+2pi).