$\cos \left({180}^{\circ }+\theta \right)$ is equal to :

$\cos \left({180}^{\circ }+\theta \right)=-\cos \left(\theta \right)$

Concept Behind the Answer:

To understand why cos(180° + θ) = -cos(θ), let’s break it down:

1. Using the Cosine Addition Formula:

The cosine of the sum of two angles is given by the formula:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

For A = 180° and B = θ, we get:

cos(180° + θ) = cos(180°)cos(θ) - sin(180°)sin(θ)

2. Substitute the Known Values for cos(180°) and sin(180°):

We know from trigonometric identities that:

cos(180°) = -1 and sin(180°) = 0

Substituting these into the formula:

cos(180° + θ) = (-1) * cos(θ) - 0 * sin(θ)

This simplifies to:

cos(180° + θ) = -cos(θ)

Therefore, cos(180° + θ) = -cos(θ), which matches option c.

Understanding the Value of cos(180°):

The value of cos(180°) is -1. In fact, cos(180°) represents the cosine of an angle on the unit circle, where the point on the circle corresponding to 180° lies on the negative x-axis, which gives a value of -1.

• cos 180° in fraction: The value of cos(180°) can be written as -1, which is a whole number but can also be expressed as -1/1 in fraction form.
• cos 180° value: The value of cos(180°) is -1, which is derived from the unit circle definition.
• cos(180° + θ): Using the cosine addition formula, we find that cos(180° + θ) = -cos(θ), meaning the cosine of an angle shifted by 180° is the negative of the original cosine value.

This is how we can change the given expression into a simplified form and understand the underlying trigonometric principles.