Chapter 9: Problem 10

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

Short Answer

Expert verified

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.

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

Short Answer

Step by step solution

Convert angles to radians

Apply cosine difference formula

Evaluate trigonometric functions

Simplify and conclude

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