Answer:
[tex]\tan\left(\dfrac{3\pi}{4}-\pi\right)=\tan\left(\dfrac{3\pi}{4}\right)=-1[/tex]
[tex]\tan\left(\dfrac{3\pi}{4}+7\pi\right)=\tan\left(\dfrac{3\pi}{4}\right)=-1[/tex]
Step-by-step explanation:
Given:
[tex]\tan\left(\dfrac{3\pi}{4}\right)=-1[/tex]
The period of a trigonometric function is the length of one complete cycle. It is the smallest positive constant distance along the x-axis over which the function's values repeat.
As the tangent function has a periodicity of π, if we add or subtract integer multiples of π from the angle, the function will have the same value:
[tex]\tan\left(\theta \pm \pi n\right)=\tan \left(\theta\right)[/tex]
Therefore, by subtracting or adding integer multiples of π to the angle of the given function (3π/4), it will have the same value as tan(3π/4):
[tex]\tan\left(\dfrac{3\pi}{4}-\pi\right)=\tan\left(\dfrac{3\pi}{4}\right)=-1[/tex]
[tex]\tan\left(\dfrac{3\pi}{4}+7\pi\right)=\tan\left(\dfrac{3\pi}{4}\right)=-1[/tex]
Answer:
[tex] \tan\left(\dfrac{3\pi}{4} - \pi\right) = -1 [/tex]
[tex] \tan\left(\dfrac{3\pi}{4} - \pi\right) = \tan\left(\dfrac{3\pi}{4} + 7\pi\right) = -1 [/tex]
Step-by-step explanation:
Let's analyze each case:
1. tan(3π/4 - π):
[tex] \tan\left(\dfrac{3\pi}{4} - \pi\right) [/tex]
Since [tex]\tan(\pi) = 0[/tex], we can rewrite the expression:
[tex] \tan\left(\dfrac{3\pi}{4} - \pi\right) = \tan\left(\dfrac{3\pi}{4}\right) [/tex]
The value of [tex]\tan(\theta)[/tex] for [tex]3\pi/4[/tex] is [tex] -1 [/tex]. Therefore,
[tex] \tan\left(\dfrac{3\pi}{4} - \pi\right) = -1 [/tex]
So, the value is the same as [tex]\tan\left(\dfrac{3\pi}{4}\right)[/tex].
2. tan(3π/4 + 7π):
[tex] \tan\left(\dfrac{3\pi}{4} + 7\pi\right) [/tex]
Since [tex]2\pi[/tex] is equivalent to a full revolution and doesn't affect the value of the tangent function, we can rewrite this expression as:
[tex] \tan\left(\dfrac{3\pi}{4} + 7\pi\right) = \tan\left(\dfrac{3\pi}{4}\right) [/tex]
As mentioned earlier, the value of [tex]\tan(\dfrac{3\pi}{4})[/tex] is [tex] -1 [/tex].
Therefore, in both cases, the values are the same:
[tex] \tan\left(\dfrac{3\pi}{4} - \pi\right) = \tan\left(\dfrac{3\pi}{4} + 7\pi\right) = -1 [/tex]
In summary, the values of [tex] \tan\left(\dfrac{3\pi}{4} - \pi\right) [/tex] and [tex] \tan\left(\dfrac{3\pi}{4} + 7\pi\right) [/tex] are both equal to [tex] -1 [/tex].
