Question

Find sin x /2 , cos x/ 2 , and tan x/ 2 from the given information. sin(x) = 4/5 , 0° < x < 90°

Answer 1

To find sin(x/2), cos(x/2), and tan(x/2) given sin(x) = 4/5 and 0° < x < 90°, we can use the half-angle formulas. sin(x/2) = √((1 - cos(x)) / 2), cos(x/2) = √((1 + cos(x)) / 2), and tan(x/2) = √2 / √5.

Step-by-step explanation:

To find sin(x/2), cos(x/2), and tan(x/2) given sin(x) = 4/5 and 0° < x < 90°, we can use the half-angle formulas.

First, let's find sin(x/2):

sin(x/2) = ± √((1 - cos(x)) / 2)

Since x is in the first quadrant, sin(x/2) is positive, so:

sin(x/2) = √((1 - cos(x)) / 2)

Plugging in sin(x) = 4/5, we can solve for cos(x) using the Pythagorean Identity:

cos^2(x) + sin^2(x) = 1

cos^2(x) + (4/5)^2 = 1

cos^2(x) + 16/25 = 1

cos^2(x) = 9/25

cos(x) = ± 3/5

Since x is in the first quadrant, cos(x) is positive, so:

cos(x) = 3/5

Now we can find sin(x/2):

sin(x/2) = √((1 - cos(x)) / 2)

sin(x/2) = √((1 - 3/5) / 2)

sin(x/2) = √(2/5)

Next, let's find cos(x/2):

cos(x/2) = ± √((1 + cos(x)) / 2)

Since x is in the first quadrant, cos(x/2) is positive, so:

cos(x/2) = √((1 + cos(x)) / 2)

Plugging in cos(x) = 3/5, we can solve for sin(x) using the Pythagorean Identity:

sin^2(x) + cos^2(x) = 1

sin^2(x) + (3/5)^2 = 1

sin^2(x) + 9/25 = 1

sin^2(x) = 16/25

sin(x) = ± 4/5

Since x is in the first quadrant, sin(x) is positive, so:

sin(x) = 4/5

Now we can find cos(x/2):

cos(x/2) = √((1 + cos(x)) / 2)

cos(x/2) = √((1 + 3/5) / 2)

cos(x/2) = √(8/10)

Finally, let's find tan(x/2):

tan(x/2) = sin(x/2) / cos(x/2)

tan(x/2) = √(2/5) / √(8/10)

tan(x/2) = √(2/5) / (√2 * √(4/5))

tan(x/2) = √(2/5) / (2 * √(1/5))

tan(x/2) = √(2/5) / (√(5/5))

tan(x/2) = √2 / √5