Expand the expression using the sum & difference identities and simplify.sin(−θ−π2)\sin\left(-\theta-\frac{\pi}{2}\right)sin(−θ−2π)

− cos ⁡ θ -\cos\theta − cos θ

Expand the expression using the sum & difference identities and simplify. sin ( − θ − π 2 ) \sin\left(-\theta-\frac{\pi}{2}\right) sin ( − θ − 2 π )

Here we're asked to expand the expression using our sum and difference identities and then simplify. Now, the expression that we're given here is the sine of negative theta minus pi over two. So let's go ahead and use our difference formula for sine in order to expand this out. Now doing that will give me sine of negative theta times the cosine of pi over two and then subtracting the cosine of negative theta times the sine of pi over two. Now the cosine of pi over two and the sine of pi over two are two values that I know. The cosine of pi over two is equal to zero and the sine of pi over two is equal to one. So here this entire term goes away because it's being multiplied by zero. Then I'm just left with a negative cosine of negative theta because that one multiplying this just leaves me with this. Now negative cosine of negative theta. We do know another identity that could help us simplify this further because the cosine of negative theta, knowing that cosine is an even function, is really just the cosine of theta. So all that I'm left with here is negative cosine of theta and this is my final answer. Thanks for watching and I'll see you in the next one.

Expand the expression using the sum & difference identities and simplify. sin ⁡ ( − θ − π 2 ) \sin\left(-\theta-\frac{\pi}{2}\right) sin ( − θ − 2 π )

− cos ⁡ θ -\cos\theta − cos θ

− sin ⁡ θ − cos ⁡ θ -\sin\theta-\cos\theta − sin θ − cos θ

− sin ⁡ θ -\sin\theta − sin θ

12. Trigonometric Functions

Trigonometric Identities

Business Calculus

12. Trigonometric Functions

Trigonometric Identities

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Multiple Choice

Expand the expression using the sum & difference identities and simplify.
$\sin\left(-\theta-\frac{\pi}{2}\right)$

$-\sin\theta-\cos\theta$

$-\sin\theta$

$-\cos\theta$

Verified step by step guidance

Step 1: Recognize that the given expression involves trigonometric functions and requires the use of sum and difference identities. The expression to simplify is sin(-θ - π/2).

Step 2: Recall the sine difference identity: sin(a - b) = sin(a)cos(b) - cos(a)sin(b). Apply this identity to sin(-θ - π/2), where a = -θ and b = π/2.

Step 3: Substitute the values into the identity: sin(-θ - π/2) = sin(-θ)cos(π/2) - cos(-θ)sin(π/2).

Step 4: Simplify using known trigonometric values: cos(π/2) = 0 and sin(π/2) = 1. This reduces the expression to sin(-θ)(0) - cos(-θ)(1).

Step 5: Simplify further using the property sin(-θ) = -sin(θ) and cos(-θ) = cos(θ). The final simplified expression becomes -cos(θ).

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