sin(2 tan-1 7/24) (that is inverse tangent)

Draw a 7 24 25 right triangle. Label the angle whose tangent is 7/24. What is its angle? What is the sin of twice that angle.

To find the value of sin(2 tan⁻¹(7/24)), we can follow these steps:

Step 1: Draw a right triangle with side lengths 7, 24, and 25.
We can do this by applying the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, we know that the hypotenuse length is 25 (since it forms a right triangle with side lengths 7 and 24). Therefore, we can label the triangle accordingly:

/|
/ |
24/ | \ 7
/ |
/___|
25

Step 2: Identify the angle whose tangent is 7/24.
The tangent of an angle, tan⁻¹, represents the inverse tangent function. It helps us find the angle whose tangent is a particular ratio of the triangle's sides. In this case, we're looking for the angle with a tangent of 7/24. Let's label it as θ.

Step 3: Calculate the value of the angle θ.
To find the angle θ, we can use the inverse tangent function, tan⁻¹, which can be found on most scientific calculators or online tools. By plugging in the ratio 7/24, we get:

θ = tan⁻¹(7/24)
θ ≈ 16.98 degrees (rounded to two decimal places)

Step 4: Calculate sin(2θ).
Now that we have the value of θ, we can find sin(2θ) by using the double-angle identity for sine:

sin(2θ) = 2sin(θ)cos(θ)

Let's calculate sin(θ) and cos(θ):

sin(θ) = opposite / hypotenuse = 7 / 25
cos(θ) = adjacent / hypotenuse = 24 / 25

Now, substitute these values into the double-angle identity:

sin(2θ) = 2 * (7/25) * (24/25)
= 336 / 625
≈ 0.5376 (rounded to four decimal places)

Therefore, sin(2 tan⁻¹(7/24)) ≈ 0.5376.