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Chapter 9: Problem 15

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

Short Answer

Expert verified
Question: Prove that \(\sin \left(360^{\circ}-\theta\right)=-\sin \theta\). Answer: By converting the equation to radians, using the co-function identity, applying the subtraction formula for sine, and simplifying the equation, we have proven the given identity is true, i.e., \(\sin \left(360^{\circ}-\theta\right)=-\sin \theta\).

Step by step solution

01

Convert the equation into radians

To work with the trigonometric functions and their properties, it's better to convert the angles from degrees to radians. Recall that \(360^{\circ} = 2\pi\) radians, thus the conversion of the given formula becomes: $$\sin \left(2\pi-\theta\right) = -\sin \theta.$$
02

Use the co-function identity

Recall the co-function identity, for sine: \(\sin(\pi - x) = \sin x\). We can write the expression \(2\pi - \theta\) as \((2\pi - \pi) - (0 + \theta) = \pi + (\pi - \theta)\). Now we can apply the co-function identity to rewrite the equation as: $$\sin \left(\pi+(\pi-\theta)\right) = -\sin \theta.$$
03

Apply the subtraction formula for sine

The subtraction formula for sine states that \(\sin(a+b)=\sin a \cos b + \cos a\sin b\). Let a = \(\pi\) and b = \((\pi-\theta)\). Then the equation becomes: $$\sin(\pi+(\pi-\theta))=\sin \pi \cos(\pi-\theta) + \cos \pi\sin(\pi-\theta).$$
04

Simplify the equation

We can simplify the expression as follow: $$\sin \pi \cos(\pi-\theta) + \cos \pi\sin(\pi-\theta) = 0 \cdot \cos(\pi-\theta) - 1 \cdot \sin(\pi-\theta)=-\sin(\pi-\theta)$$ Now, using the co-function identity once again, which states that \(\sin(\pi - x) = \sin x\), we can see: $$-\sin(\pi-\theta)=-\sin \theta.$$ This confirms the given identity. Thus, we have successfully shown that \(\sin \left(360^{\circ}-\theta\right)=-\sin \theta\).

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Most popular questions from this chapter

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) \(\tan y=-0.3006,0^{\circ} \leq y \leq 360^{\circ}\)

If \(\sin \phi<0\) and \(\cos \phi>0\), state the quadrant in which \(\phi\) lies.

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

Show \(\sin 3 A=3 \sin A \cos ^{2} A-\sin ^{3} A\)

Evaluate (a) \(\operatorname{cosec} 37^{\circ}\) (b) \(\cot 1.3\) (c) \(\sec 40^{\circ}\)
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