can you correct the rest for me please?

Express each as a function of theta:

a. sin (270deg + theta)= cos theta
b. cos (pi + theta)= -cos theta
c. tan (810 + theta)= ?????
d. sin (theta - 180)= -sin theta

a) I get -cos theta

b) -cos theta yes

c) 810 = 360*2 + 90
so really tan (90+theta) =-cotan (theta)

d) -sin theta is correct

Sure! Let's go through each expression and find the correct function of theta:

a. To express sin (270° + theta) as a function of theta, we start by converting 270° to radians. One full rotation in radians is equal to 2π, so 270° is equal to (270/360) * 2π = (3/4) * 2π = (3/2)π.

Now we can rewrite the expression as sin((3/2)π + theta). Since sin(x + 2π) = sin(x) for any value of x, we can subtract 2π from (3/2)π to simplify it.

(3/2)π - 2π = (3/2 - 4/2)π = (-1/2)π = -π/2

Therefore, the function of theta is sin(-π/2 + theta).

b. Similar to the previous example, we convert pi (π) to degrees. Pi is equal to (180/π) degrees. So, the expression is cos(180° + theta) = cos((180/π) * π + theta) = cos(π + theta).

Using the angle addition identity for cosine, cos(x + π) = -cos(x), we can rewrite the expression as -cos(theta). Therefore, the function of theta is -cos(theta).

c. To express tan (810° + theta) as a function of theta, we first convert 810° to radians. Using the fact that one full rotation is equal to 2π radians, we have (810/360) * 2π = (9/4) * 2π = (9/2)π.

Since tan(x + π) = tan(x) for any value of x, we can subtract π from (9/2)π to simplify it.

(9/2)π - π = (9/2 - 2/2)π = (7/2)π

Therefore, the function of theta is tan((7/2)π + theta).

d. Lastly, to express sin (theta - 180°) as a function of theta, we can directly use the angle subtraction identity for sine:

sin(x - π) = -sin(x)

By substituting theta as x, the expression becomes -sin(theta). Therefore, the function of theta is -sin(theta).

I hope this helps! Let me know if you have any further questions.