find the exact value of the given expression. Thank you!

tan(1/2cos^-1(9/10)

tangent of 1/2 the angle whose cosine is 9/10

Exact value? What is wrong with a calculator? Paste this into your google search window...

tan(.5*arccos(9/10))=

Is that exact enough? It appears to be an irrational number.

find the exact value of (5pie/8)

To find the exact value of the expression tan(1/2cos^(-1)(9/10)), we can use the double-angle identity for tangent:

tan(2θ) = (2tan(θ))/(1-tan^2(θ))

Let's start by finding the value of θ.

Given: cos^(-1)(9/10)

To find the value of cos^(-1)(9/10), we need to find the angle whose cosine is 9/10.

Using the inverse cosine function, we can write:

cos^(-1)(9/10) = θ

This means that the cosine of the angle θ is equal to 9/10.

cos(θ) = 9/10

Using the Pythagorean identity, sin^2(θ) + cos^2(θ) = 1, we can find the value of sin(θ).

sin^2(θ) + cos^2(θ) = 1
sin^2(θ) + (9/10)^2 = 1
sin^2(θ) + 81/100 = 1
sin^2(θ) = 1 - 81/100
sin^2(θ) = 19/100
sin(θ) = √(19/100) = √19/10

Now, we have the values of cos(θ) and sin(θ), and we can find the value of tan(θ).

tan(θ) = sin(θ)/cos(θ)
tan(θ) = (√19/10)/(9/10)
tan(θ) = √19/9

Now, let's find the exact value of tan(1/2θ):

tan(1/2θ) = (2tan(θ))/(1-tan^2(θ))
tan(1/2θ) = (2*(√19/9))/(1-(√19/9)^2)
tan(1/2θ) = (2√19/9)/(1-19/81)
tan(1/2θ) = (2√19/9)/(81-19)
tan(1/2θ) = (2√19/9)/(62)
tan(1/2θ) = 2√19/558

So, the exact value of the expression tan(1/2cos^(-1)(9/10)) is 2√19/558.

To find the exact value of the given expression, we'll follow these steps one by one:

Step 1: Start with the innermost function, `cos^-1(9/10)`. This represents the inverse cosine function of 9/10.

Step 2: Use the definition of the inverse cosine function, which states that `cos^(-1)(x) = arccos(x)`. So, `cos^-1(9/10)` is equivalent to `arccos(9/10)`.

Step 3: Calculate the value of `arccos(9/10)`. To do this, we need to find the angle whose cosine is 9/10.

Step 4: Using the Pythagorean identity `sin^2(x) + cos^2(x) = 1`, we can determine the sine value. Since cosine is 9/10, we can solve for sine: `sin^2(x) + (9/10)^2 = 1`.

Step 5: Simplify the equation: `sin^2(x) + 81/100 = 1`. Subtracting 81/100 from both sides, we get `sin^2(x) = 19/100`.

Step 6: Take the square root of both sides: `sin(x) = √(19/100)`. Simplifying further, `sin(x) = √19/10`.

Step 7: Convert the value of sine to tangent, since that's what we're ultimately looking for: `tan(x) = sin(x) / cos(x)`. Plugging in the values we have, `tan(x) = (√19/10) / (9/10)`. Simplify by canceling out the 10s, `tan(x) = √19/9`.

So, the exact value of the expression `tan(1/2cos^-1(9/10))` is `√19/9`.