find (s+t) given that cos s=1/3 with s in quad I and sin T=-1/2 with t in quad IV

I have a feeling that you want exact values for

sin(x+t) or cos(s+t)

the way it stands, just use your calculator to find each of the angles, then add them up.

No it is to find the exact value of the expression using the provided information. Find Tan (S+T) given that cos s=1/3 with s in quad I and sin T= -1/2 with T in quad IV

To find (s + t), we need to determine the values of both s and t. We are given that cos s = 1/3 and sin t = -1/2.

Let's begin by finding s. We are given that cos s = 1/3. Since s is in Quadrant I, where cosine is positive, we can use the Pythagorean identity to find the value of sin s:

sin^2 s + cos^2 s = 1

(1/3)^2 + sin^2 s = 1

1/9 + sin^2 s = 1

sin^2 s = 1 - 1/9

sin^2 s = 8/9

Taking the square root of both sides, we get:

sin s = ± sqrt(8/9)

Since s is in Quadrant I, sin s must be positive. Therefore, sin s = sqrt(8/9).

Now let's find t. We are given that sin t = -1/2. Since t is in Quadrant IV, where sine is negative, we can use the Pythagorean identity to find the value of cos t:

sin^2 t + cos^2 t = 1

(-1/2)^2 + cos^2 t = 1

1/4 + cos^2 t = 1

cos^2 t = 1 - 1/4

cos^2 t = 3/4

Taking the square root of both sides, we get:

cos t = ± sqrt(3/4)

Since t is in Quadrant IV, cos t must be positive. Therefore, cos t = sqrt(3/4).

Now that we have determined the values of sin s, sin t, cos s, and cos t, we can find (s + t):

(s + t) = arcsin(sqrt(8/9)) + arccos(sqrt(3/4))

To find the values of arcsin(sqrt(8/9)) and arccos(sqrt(3/4)), you can use a scientific calculator or a trigonometric table. Plug in the values and evaluate the expression to find the sum (s + t).