Chapter 9: Problem 13

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

Short Answer

Expert verified

Question: Prove the trigonometric identity \(\cos(180^{\circ}+\theta)=-\cos \theta\). Answer: By applying the sum of angles formula and simplifying the result, we proved that \(\cos(180^{\circ}+\theta)=-\cos \theta\).

Step by step solution

Use the Sum of Angles formula

Let's take the left-hand side of the equation and use the formula for the cosine of the sum of two angles. According to the formula, \(\cos(A + B) = \cos A \cos B - \sin A \sin B.\) Replacing A with \(180^{\circ}\) and B with \(\theta\), we get: \(\cos \left(180^{\circ}+\theta\right)=\cos 180^{\circ} \cos \theta - \sin 180^{\circ} \sin \theta\)

Determine the value of cosine and sine

The cosine of \(180^{\circ}\) is \(-1\) and the sine of \(180^{\circ}\) is \(0.\) Substitute these values in the equation: \(\cos(180^{\circ}+\theta)= -1 \cdot \cos \theta - 0 \cdot \sin \theta\)

Simplify the result

Simplifying the expression obtained, we get: \(\cos(180^{\circ}+\theta)=-\cos \theta - 0\), which simplifies to \(\cos(180^{\circ}+\theta)=-\cos \theta\)

Compare with the right-hand side

Now, observe that left-hand side is exactly equal to the right-hand side. This implies that the given trigonometric identity is correct. Therefore, we have successfully proved the identity \(\cos(180^{\circ}+\theta)=-\cos \theta\) using the sum of angles formula in trigonometry.

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

Short Answer

Step by step solution

Use the Sum of Angles formula

Determine the value of cosine and sine

Simplify the result

Compare with the right-hand side

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