Question

If sin(t) = 18 and t is in quadrant I, find the exact value of sin(2t), cos(2t), and tan(2t) algebraically without solving for t.

Answer 1

To find the exact values of sin(2t), cos(2t), and tan(2t) when sin(t) = 18 and t is in quadrant I, we can use trigonometric identities.

Step-by-step explanation:

To find the exact values of sin(2t), cos(2t), and tan(2t) when sin(t) = 18 and t is in quadrant I, we can use trigonometric identities. Let's start with sin(2t):

1. Using the double angle formula for sin, sin(2t) = 2sin(t)cos(t)
2. Let's substitute sin(t) = 18: sin(2t) = 2(18)cos(t)
3. Since t is in quadrant I, cos(t) is positive, so we can use the Pythagorean identity sin^2(t) + cos^2(t) = 1 to find cos(t): cos(t) = sqrt(1 - sin^2(t)) = sqrt(1 - 18^2)
4. Substitute cos(t) into the equation and simplify to find the exact value of sin(2t)

Next, let's find the exact values of cos(2t) and tan(2t) using similar steps.