Problem 63 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \sin \left(-300^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value of \(\sin(-300^\circ)\) is \(-\sqrt{3}/2\).

Step 1: Breaking down the angle

First break down the \(-300^\circ\) angle into two angles that are easier to deal with. It can be express as a difference of \(-360^\circ +60^\circ \). This step simplifies the calculation by using angles that are familiar from the unit circle.

Step 2: Use sum or difference identity

Secondly, apply the sum or difference identity for sine. The difference identity for sine is given by: \(\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)\). In this case, \(a = -360^\circ\) and \(b = 60^\circ\). So it becomes \(\sin(-360^\circ)\cos(60^\circ) - \cos(-360^\circ)\sin(60^\circ)\). We can now substitute the known values for these sines and cosines from the unit circle.

Step 3: Substitute known values

The third step is substituting the values. We know from the unit circle that \(\sin(-360^\circ)=0\), \(\cos(60^\circ)=0.5\), \(\cos(-360^\circ)=1\), and \(\sin(60^\circ)=\sqrt{3}/2\). Therefore, the expression becomes \(0*0.5 - 1*(\sqrt{3}/2) = -(\sqrt{3}/2)\).

Key Concepts

Angle Simplification

Simplifying angles can make trigonometric calculations much easier, especially if angle values fall outside common reference angles. An important aspect is reducing the given angle to one that is more recognizable using known properties. For example, consider the angle \(-300° \). It is easier if we recognize it can be rewritten as \(-360° + 60°\). This simplification reduces a complex angle into two parts: one full rotation and a manageable angle of \(60°\). This step essentially aligns the problem into simpler terms using reference angles. By reducing a complex angle to its components, calculations become more manageable, highlighting the beauty and relevance of angle simplification.

Trigonometric Identities

Trigonometric identities are tools that help solve problems involving trigonometric values. They allow us to manipulate and simplify expressions efficiently. When dealing with angle simplification, the sum and difference identities come in handy. The difference identity for sine is particularly useful here. It states:

• \(\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)\)

These identities are fundamental because they provide a structured way to find the sine, cosine, or tangent of non-standard angles. By using these identities, we find solutions quicker without relying solely on the unit circle angles. This identity acts like a recipe, where once you plug in the known values, it simplifies the task of calculating the trigonometric function of compound angles.

Unit Circle Values

The unit circle is an essential tool in trigonometry that places trigonometric functions into perspective. It is a circle with radius 1 centered at the origin of a coordinate system. The sine, cosine, and tangent functions for angles are easily derived from this circle. When simplifying angles, such as \(-360° + 60°\), knowing unit circle values is crucial. For instance:

• \(\sin(-360°) = 0\)
• \(\cos(60°) = 0.5\)
• \(\cos(-360°) = 1\)
• \(\sin(60°) = \sqrt{3}/2\)

These values allow you to seamlessly substitute and simplify calculations. By memorizing or understanding these values on the unit circle, you can flawlessly compute equations such as sine or cosine using sum or difference identities. The unit circle bridges the knowledge of angle simplification and trigonometric identities into executable operations.