Question

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

Short answer

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

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: $$\sin \left(2\pi-\theta\right) = -\sin \theta.$$

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: $$\sin \left(\pi+(\pi-\theta)\right) = -\sin \theta.$$

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: $$\sin(\pi+(\pi-\theta))=\sin \pi \cos(\pi-\theta) + \cos \pi\sin(\pi-\theta).$$

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

Key concepts

Radian Conversion

When working with trigonometric functions, it is often more convenient to use radians rather than degrees. This is because most calculus operations and trigonometric identities are written in radians.
To convert degrees to radians, you use the conversion factor \(\frac{\pi}{180}\). Therefore, 360 degrees is equivalent to \(2\pi\) radians. This means that a full circle in radians is \(2\pi\).

• Radian is a measure of angle based on the radius of a circle.
• It provides a natural way to describe rotation and circular motion.
• Using radians, many trigonometric patterns and equations simplify, as seen in our exercise.

In our given problem, converting \(360^\circ\) to radians results in \(2\pi\), thus the equation \(\sin(360^\circ - \theta) = -\sin \theta\) becomes \(\sin(2\pi - \theta) = -\sin \theta\). This is the first important step, allowing us to engage with more complex trigonometric tools.

Sine Function

The sine function is a fundamental trigonometric function, representing the y-coordinate of a point on the unit circle as it relates to an angle's rotation from the positive x-axis. The unit circle has a radius of 1.

• \(\sin \theta\) is the opposite side over hypotenuse in a right triangle.
• When using the unit circle, \(\sin \theta\) is simply the y-coordinate of the endpoint of the radius.
• The sine function has periodicity; it repeats every \(2\pi\) radians or \(360^\circ\).

An important property, when working with the sine function, is understanding how it behaves under angle transformations such as reflections. For instance, \(\sin(\pi + x) = -\sin x\). This property helps explain the behavior when angles move beyond the initial quadrants on the unit circle. In this problem, we use the negative sine identity, \(\sin(\pi-x) = \sin x\), and extend that understanding to show how sine changes with \(360^\circ - \theta\).

Angle Subtraction Identity

Keeping up with our scenario, understanding angle subtraction identities is crucial because it allows us to simplify and solve trigonometric equations involving sums and differences of angles. The subtraction formula for sine is: \(\sin(a - b) = \sin a \cos b - \cos a \sin b\).

• This identity helps in breaking down complicated angles into simpler components.
• In the exercise, it assists in deconstructing \(\sin(2\pi - \theta)\).
• The framework provides conversion of operations: subtraction in angles converts into addition in sine identities.

Applying this identity, we see how \(\sin(\pi + (\pi - \theta))\) translates to an expression involving cosine and sine multiplications.

The flexibility of this identity lies in its ability to transform sums and differences into easily computable parts by utilizing familiar cosine and sine values, leading to simplifications, as demonstrated with \(-\sin(\pi - \theta) = -\sin \theta\). Understanding these transformations builds foundational skills for tackling broader trigonometric challenges.