Question

If sin t =

2/3
and t is in quadrant I find the exact value of sin (2 t), cos (2 t), and
tan (2 t) algebraically without solving for t.
Enclose numerators and denominators in parentheses. For example, (a - b)/(1+n).

Answer 1

sin(2t) = 4√(5)/9, cos(2t) = 1/9, tan(2t) = 12/5

Step-by-step explanation:

To find the exact value of sin(2t), cos(2t), and tan(2t) when sin(t) = 2/3 and t is in quadrant I, we can use the double-angle identities for sine, cosine, and tangent.

1. The double-angle identity for sine is sin(2t) = 2sin(t)cos(t). Using the given value of sin(t) = 2/3, we can substitute the values into the formula to get sin(2t) = 2(2/3)(√(1 - (2/3)^2)) = 4√(5)/9.
2. The double-angle identity for cosine is cos(2t) = cos^2(t) - sin^2(t). Using the given value of sin(t) = 2/3, we can find cos(t) = √(1 - (2/3)^2) = √(5)/3. Substituting these values into the formula, we get cos(2t) = (√(5)/3)^2 - (2/3)^2 = 5/9 - 4/9 = 1/9.
3. The double-angle identity for tangent is tan(2t) = 2tan(t) / (1 - tan^2(t)). Using the given value of sin(t) = 2/3, we can find cos(t) = √(1 - (2/3)^2) = √(5)/3. Substituting these values, we get tan(2t) = 2(2/3) / (1 - (2/3)^2) = 4/3 / (1 - 4/9) = 4/3 × 9/5 = 12/5.