Question

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

Short answer

Question: Prove that \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\). Answer: Using trigonometric identities and converting angles to radians, we applied the cosine difference formula, evaluated the trigonometric functions, and simplified the expression to show that \(\cos \left(180^{\circ}-\theta\right) = -\cos \theta\).

Step-by-step solution

Convert angles to radians

To make the calculations easier, we will convert the angles from degrees to radians. In general, to convert an angle from degrees to radians, we need to use the formula \(\text{radians} = \frac{\text{degrees} \times \pi}{180}\). In this case, we will convert both \(180^{\circ}\) and \(\theta\) to radians. So, let's replace \(180^{\circ}\) with \(\pi\) and rewrite the expression as: \(\cos \left(\pi - \theta\right)=-\cos \theta\)

Apply cosine difference formula

We can use the cosine difference formula, which states that \(\cos (a - b) = \cos a\cos b + \sin a\sin b\). In our case, \(a = \pi\) and \(b = \theta\). So let's apply the formula to our expression: \(\cos \left(\pi - \theta\right) = \cos \pi\cos \theta + \sin \pi\sin \theta\)

Evaluate trigonometric functions

Now, we need to evaluate \(\cos \pi\) and \(\sin \pi\). We know that \(\cos \pi = -1\) and \(\sin \pi = 0\). So we can substitute these values into our expression: \(-1\cos \theta + 0\sin \theta = -\cos \theta\)

Simplify and conclude

Since \(0\sin \theta = 0\), the expression simplifies to: \(- \cos \theta = -\cos \theta\) This indicates that the statement \(\cos \left(180^{\circ}-\theta\right) = -\cos \theta\) is true, concluding the problem.

Key concepts

Radian Conversion

When working with trigonometric expressions, it is common to convert angles from degrees to radians because radians are the standard unit of angular measure in mathematics. This conversion simplifies computations and aligns with most mathematical formulas. To convert an angle from degrees to radians, use the formula:

• radians = \( \frac{\text{degrees} \times \pi}{180} \)

For example, to convert \(180^{\circ}\) to radians, multiply:

• \( \frac{180 \times \pi}{180} = \pi \)

Thus, this formula tells us that \(180^{\circ}\) is equivalent to \(\pi\) radians. Whenever you encounter angles in degrees within a trigonometric context, consider converting them to radians for a smoother workflow.

Cosine Difference Formula

The cosine difference formula is a useful trigonometric identity that helps simplify expressions involving the cosine of a difference of two angles. It is written as:

• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

This formula is derived from the properties of trigonometric functions on the unit circle. In our problem, we apply it by setting \(a = \pi\) and \(b = \theta\). Applying the formula:

• \( \cos(\pi - \theta) = \cos \pi \cos \theta + \sin \pi \sin \theta \)

Using specific angle values, the formula becomes a reliable tool for breaking down more complex trigonometric expressions.

Trigonometric Evaluations

Once you've expressed your problem using a trigonometric identity, the next step is usually to evaluate the trigonometric functions involved. This involves calculating values like \( \cos \pi \) and \( \sin \pi \). These are angles known from the unit circle:

• \( \cos \pi = -1 \)
• \( \sin \pi = 0 \)

Substituting these values into your expression makes further simplification possible. For example:

• \( \cos(\pi - \theta) = -1 \cdot \cos \theta + 0 \cdot \sin \theta \)

This substitution simplifies the problem, confirming the original expression with concrete values.

Cosine Properties

Cosine, like its sibling sine, possesses certain properties that simplify the way we approach mathematical problems. Some salient properties include:

• Symmetry: Cosine is an even function, meaning \( \cos(-\theta) = \cos \theta \).
• Periodicity: The cosine function is periodic with a period of \(2\pi\), meaning every \(2\pi\) radians its values repeat.

In this exercise, another vital property is used:

• For angles of the form \( \pi + \theta \) or \( \pi - \theta \), cosine gives opposite results to \( \cos\theta \) when expressed as an identity: \( \cos(\pi - \theta) = -\cos \theta \).

This characteristic is key in proving the given problem as it directly relates to the defined properties of the cosine function.