Final answer:
To find cos(t/2) given tan(t) = 4/3, use the half-angle identity for cosine.
Step-by-step explanation:
If t is in a quadrant where tan t is positive and tan t = 4/3, we need to find cos t/2. First, using the identity tan t = sin t / cos t, we can set up a right-angled triangle with the opposite side being 4 units and the adjacent side being 3 units. Using the Pythagorean theorem, the hypotenuse will be √(4^2+3^2) = √(16+9) = √25 = 5 units. Therefore, sin t = 4/5 and cos t = 3/5.
To find cos t/2, we use the half-angle formulas. The relevant formula is cos t/2 = ± √((1+cos t)/2), and the sign depends on the quadrant where t/2 lies. Since we are not provided with the exact quadrant of t, we cannot precisely determine the sign of cos t/2, but we can provide the absolute value: cos t/2 = √((1+(3/5))/2) = √(1.6/2) = √(0.8).
To find cos(t/2) given that tan(t) = 4/3, we can use a trigonometric identity. First, we need to find the value of cos(t). Since tan(t) = 4/3, we can use the Pythagorean identity tan(t) = sin(t) / cos(t) to solve for cos(t).
sin(t) = 4 and cos(t) = 3. Now, we can use the half-angle identity for cosine: cos(t/2) = sqrt((1 + cos(t)) / 2). Plugging in the value of cos(t) in the formula, we get:
cos(t/2) = sqrt((1 + 3) / 2)
cos(t/2) = sqrt(4 / 2) = sqrt(2)