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

Certainly! I can help you with that.

a. To express sin(270° + theta) as a function of theta, we can use the angle sum formula for sine: sin(a + b) = sin(a)cos(b) + cos(a)sin(b). In this case, let a = 270° and b = theta. Therefore, sin(270° + theta) = sin(270°)cos(theta) + cos(270°)sin(theta).
Now, we know that sin(270°) = -1 and cos(270°) = 0. Substituting these values, we get sin(270° + theta) = -1cos(theta) + 0sin(theta) = -cos(theta).

b. Similarly, to express cos(pi + theta) as a function of theta, we can use the angle sum formula for cosine: cos(a + b) = cos(a)cos(b) - sin(a)sin(b). In this case, let a = pi and b = theta. Therefore, cos(pi + theta) = cos(pi)cos(theta) - sin(pi)sin(theta).
Now, we know that cos(pi) = -1 and sin(pi) = 0. Substituting these values, we get cos(pi + theta) = -1cos(theta) - 0sin(theta) = -cos(theta).

c. To express tan(810 + theta) as a function of theta, we can use the angle sum formula for tangent: tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)). In this case, let a = 810 and b = theta. Therefore, tan(810 + theta) = (tan(810) + tan(theta)) / (1 - tan(810)tan(theta)).
However, the value of tan(810) is not a commonly used value, so we don't have a direct expression in terms of theta for this case.

d. To express sin(theta - 180) as a function of theta, we can use the angle difference formula for sine: sin(a - b) = sin(a)cos(b) - cos(a)sin(b). In this case, let a = theta and b = 180°. Therefore, sin(theta - 180) = sin(theta)cos(180°) - cos(theta)sin(180°).
Now, we know that cos(180°) = -1 and sin(180°) = 0. Substituting these values, we get sin(theta - 180) = sin(theta)(-1) - cos(theta)(0) = -sin(theta).

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