Problem 37 - Algebra 2

Question

Verify that each of the following is an identity. \(\sin \left(60^{\circ}+\theta\right)+\sin \left(60^{\circ}-\theta\right)=\sqrt{3} \cos \theta\)

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\sqrt{3} \cos \theta\).

Step 1: Use Angle-Sum and Angle-Difference Formulas

To verify the identity, start by applying the angle-sum formula: \[\sin(a + b) = \sin a \cos b + \cos a \sin b\] for \(\sin(60^{\circ} + \theta)\) and the angle-difference formula: \[\sin(a - b) = \sin a \cos b - \cos a \sin b\] for \(\sin(60^{\circ} - \theta)\).

Step 2: Expand the Sinusoidal Terms

Apply the formulas to expand both terms: \[\sin(60^{\circ} + \theta) = \sin 60^{\circ} \cos \theta + \cos 60^{\circ} \sin \theta\] \[\sin(60^{\circ} - \theta) = \sin 60^{\circ} \cos \theta - \cos 60^{\circ} \sin \theta\] Recall that \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) and \(\cos 60^{\circ} = \frac{1}{2}\).

Step 3: Combine Like Terms

Substitute the known values and combine the expressions: \[(\frac{\sqrt{3}}{2} \cos \theta + \frac{1}{2} \sin \theta) + (\frac{\sqrt{3}}{2} \cos \theta - \frac{1}{2} \sin \theta)\] This simplifies to: \[\frac{\sqrt{3}}{2} \cos \theta + \frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta - \frac{1}{2} \sin \theta = \sqrt{3} \cos \theta\] The \(\sin \theta\) terms cancel out.

Step 4: Verify the Final Result

After cancellation, you're left with: \[\sqrt{3} \cos \theta\] This is equivalent to the right side of the original identity. Therefore, the original equation is indeed an identity.

Key Concepts

Angle-Sum Formula

The Angle-Sum Formula is a vital tool in trigonometry, particularly when dealing with compound angles like \(\sin(60^{\circ} + \theta)\).
This formula reveals how to break down the sine of an angle sum into a sum of products of sine and cosine of individual angles.
The general form is:

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

In this formula, you expand the angle sum into two separate trigonometric operations. Remembering this structure is crucial for solving complex trigonometric identities efficiently, such as the one given in your exercise.

Angle-Difference Formula

Similarly to the Angle-Sum Formula, the Angle-Difference Formula is used to handle expressions like \(\sin(60^{\circ} - \theta)\).
It allows the decomposition of the sine of an angle difference into various trigonometric operations.
The formula is:

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

This formula is particularly handy when simplifying expressions involving differences of angles. In this specific identity, it simplifies the components under subtraction elegantly, aiding in the cancellation of terms as seen in your solution.

Trigonometric Functions

Trigonometric functions, especially sine and cosine, are the backbone of this exercise. Understanding how these functions behave with different angles is key.
Sine and cosine relate a given angle of a triangle to ratios of side lengths. When decomposing compounded angles, you're essentially redistributing these ratios.
They follow periodic cycles, making them predictable and calculating angles easier. These inherent properties of sine and cosine allow us to simplify equations such as:

• \(\sin(60^{\circ} + \theta)\) and \(\sin(60^{\circ} - \theta)\)

Trigonometric functions are indispensable when working with identities, as they aid in converting complex expressions to simpler forms.

Sin and Cos Values

Knowledge of specific sine and cosine values at common angles is immensely helpful for verifying equations such as this one.
Take, for instance, \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) and \(\cos 60^{\circ} = \frac{1}{2}\).
These values are derived from equilateral triangle properties or unit circle coordinates. Knowing these fixed values quickly confirms parts of the expanded expressions during verification.
Through practice, you can memorize these values, which in turn streamlines your ability to simplify and solve trigonometric identities efficiently.