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Chapter 9: Problem 12

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

Short Answer

Expert verified
Answer: The value of \(\sin(180^{\circ}+\theta)\) is \(-\sin\theta\).

Step by step solution

01

Convert Degrees to Radians

To work with the angle addition formula for sine, we need to convert the given angle from degrees to radians. We can do this by multiplying the angle in degrees by the conversion factor \(\frac{\pi}{180}\). $$ 180^{\circ} = 180 \cdot \frac{\pi}{180} = \pi $$ Now we can rewrite the given expression as: $$ \sin(\pi + \theta) $$
02

Apply the Angle Addition Formula for Sine

To solve for \(\sin(\pi + \theta)\), we need to use the angle addition formula for sine, which states: $$ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) $$ In our case, \(\alpha = \pi\) and \(\beta = \theta\). Let's plug these values into the formula: $$ \sin(\pi + \theta) = \sin(\pi)\cos(\theta) + \cos(\pi)\sin(\theta) $$
03

Use Trigonometric Properties

Now we can use some basic trigonometric properties to simplify the expression further. Recall that \(\sin(\pi) = 0\) and \(\cos(\pi) = -1\). This allows us to simplify the expression as follows: $$ \sin(\pi + \theta) = 0 \cdot \cos(\theta) + (-1) \cdot \sin(\theta) = -\sin(\theta) $$ Now, we have the final result: $$ \sin(180^{\circ}+\theta) = -\sin\theta $$

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Most popular questions from this chapter

Convert the following angles in degrees to radians: (a) \(12^{\circ}\) (b) \(65^{\circ}\) (c) \(200^{\circ}\) (d) \(340^{\circ}\) (e) \(1000^{\circ}\)

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

Convert the following angles in radians to degrees: (a) \(0.3609\) (b) \(0.4771\) (c) \(1.3692\) (d) \(\frac{\pi}{3}\) (e) \(\frac{2 \pi}{3}\) (f) \(6 \pi\) (g) \(\frac{\pi}{5}\) (h) \(\frac{3 \pi}{2}\)

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

Show $$ \frac{\sin 3 A}{\sin 2 A}=2 \cos A-\frac{1}{2 \cos A} $$
See all solutions

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