Question

Which expression does sine of the quantity pi plus theta end quantity plus cosine of the quantity pi over 2 minus theta end quantity simplify to?

sin θ + cos θ
2cos θ
0
1

(Equation in photo below)

Answer 1

(c) 0

Explanation:

Each of the terms in the expression represents a different transformation of a different trig function. Expressing those as the same trig function can make it easier to find the sum.

__

We can start with the identity ...

cos(x) = sin(x +π/2)

Substituting the argument of the cosine function in the given expression, we have ...

cos(π/2 -θ) = sin((π/2 -θ) +π/2) = sin(π -θ) = -sin(θ -π)

__

The first term, sin(π +θ), is a left-shift of the sine function by 1/2 cycle, so can be written ...

sin(π +θ) = -sin(θ)

The second term is the opposite of a right-shift of the sine function by 1/2 cycle, so can be written ...

cos(π/2 -θ) = -sin(θ -π) = sin(θ)

Then the sum of terms is ...

sin(π +θ) +cos(π/2 -θ) = -sin(θ) +sin(θ) = 0

The sum of the two terms is identically zero.