Please, rewrite the expression as an algebraic expression in x. sin(tan^

tinfoQ

tinfoQ

Answered question

2021-08-22

Please, rewrite the expression as an algebraic expression in x.
sin(tan1x)

Answer & Explanation

sweererlirumeX

sweererlirumeX

Skilled2021-08-23Added 91 answers

Let θ=tan1(x), then tanθ=x
Draw a right triangle with θ as one of it’s acute angles:
1) AC=AB2+DC2
AC=x2+12=1+x2
2) sin(tan1(x))=x1+x2

karton

karton

Expert2023-06-11Added 613 answers

Step 1: Recall the identity tan1x=arctan(x).
Step 2: Use the definition of the tangent function: tan(θ)=sin(θ)cos(θ). In this case, θ=tan1x. So we can rewrite tan1x as arctan(x)=sin(arctan(x))cos(arctan(x)).
Step 3: Apply the identity sin(arctan(x))=x1+x2. This can be derived using the right triangle definition of the trigonometric functions.
Step 4: Apply the identity cos(arctan(x))=11+x2. Again, this can be derived using the right triangle definition of the trigonometric functions.
Step 5: Substitute the values of sin(arctan(x)) and cos(arctan(x)) back into the expression sin(arctan(x))cos(arctan(x)).
Combining all the steps, we get:
sin(tan1x)=sin(sin(arctan(x))cos(arctan(x))) =sin(arctan(x))cos(arctan(x)) =x1+x211+x2 =x1+x2·1+x2 =x·1+x21+x2 =x
Therefore, the expression sin(tan1x) can be rewritten as the algebraic expression x.
star233

star233

Skilled2023-06-11Added 403 answers

To rewrite the expression sin(tan1x) as an algebraic expression in x, we can use trigonometric identities. First, we'll start by using the identity tan1x=arctan(x):
sin(tan1x)=sin(arctan(x))
Next, we can use the identity sin(arctan(x))=x1+x2:
sin(tan1x)=sin(arctan(x))=x1+x2
Therefore, the algebraic expression in x equivalent to sin(tan1x) is x1+x2.
user_27qwe

user_27qwe

Skilled2023-06-11Added 375 answers

Answer:
x1+x2
Explanation:
tan1x=arctanx
Using this identity, we have:
sin(tan1x)=sin(arctanx)
Now, let's use another trigonometric identity to express sin(arctanx) in terms of x. The identity is:
sin(arctanx)=x1+x2
Therefore, the algebraic expression in x equivalent to sin(tan1x) is:
x1+x2

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