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Chapter 9: Problem 16

Show \(\cos \left(360^{\circ}-\theta\right)=\cos \theta\)

Short Answer

Expert verified
Question: Prove that the cosine of an angle difference (360° - θ) is equal to the cosine of the original angle (θ). Answer: Based on our calculations, we proved that \(\cos(360^{\circ} - \theta)=\cos(\theta)\) using the angle difference formula for cosine.

Step by step solution

01

Write the angle difference formula for cosine

Recall that the angle difference formula for cosine is: \(\cos(a - b)=\cos a \cos b + \sin a \sin b\).
02

Substitute the given values into the formula

Let's substitute \(a=360^{\circ}\) and \(b=\theta\) into the formula: \(\cos(360^{\circ} - \theta)=\cos(360^{\circ})\cos(\theta) + \sin(360^{\circ})\sin(\theta)\).
03

Evaluate the cosine and sine of 360°

We know that \(\cos(360^{\circ})=1\) and \(\sin(360^{\circ})=0\). Let's substitute these values into the expression: \(\cos(360^{\circ} - \theta)=1\cdot\cos(\theta) + 0\cdot\sin(\theta)\).
04

Simplify the expression

Now, we simplify the expression: \(\cos(360^{\circ} - \theta)=\cos(\theta)\). Therefore, we have demonstrated that \(\cos(360^{\circ} - \theta)=\cos(\theta)\).

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Most popular questions from this chapter

Show \(\sin \left(\theta+\frac{\pi}{2}\right)=\cos \theta\)

Simplify $$ \sin A \cos A \tan A+\frac{2 \sin A \cos ^{3} A}{\sin 2 A} $$

Solve (a) \(\sin \theta=0.3510,0^{\circ} \leq \theta \leq 360^{\circ}\) (b) \(\sin \theta=0.4161,0 \leq \theta \leq 2 \pi\) (c) \(\cos t=-0.3778,0 \leq t \leq 2 \pi\) (d) \(\cos x=0.7654,0^{\circ} \leq x \leq 360^{\circ}\) (e) \(\tan y=1.7136,0^{\circ} \leq y \leq 360^{\circ}\) (f) \(\tan y=-0.3006,0^{\circ} \leq y \leq 360^{\circ}\)

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

A voltage source, \(v(t)\), varies with time, \(t\), according to $$ v(t)=50 \sin (\pi t+10) $$ State (a) the angular frequency, (b) the phase, (c) the amplitude, (d) the period, (e) the time displacement, (f) the frequency of the voltage.
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