Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\sin \left(210^{\circ}+x\right)-\cos \left(120^{\circ}+x\right)=0$$

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to 0.

Step 1: Simplify the Sine Term

We begin by using the angle addition formula for sine: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B. \] Applying this to \( \sin(210^{\circ} + x) \), we recognize that \( \sin 210^{\circ} = -\frac{1}{2} \) and \( \cos 210^{\circ} = -\frac{\sqrt{3}}{2} \). Thus, \[ \sin(210^{\circ} + x) = \left(-\frac{1}{2}\right)\cos x + \left(-\frac{\sqrt{3}}{2}\right)\sin x = -\frac{1}{2}\cos x - \frac{\sqrt{3}}{2}\sin x. \]

Step 2: Simplify the Cosine Term

Next, use the angle addition formula for cosine: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B. \] Substitute into \( \cos(120^{\circ} + x) \). Knowing \( \cos 120^{\circ} = -\frac{1}{2} \) and \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \), we have: \[ \cos(120^{\circ} + x) = \left(-\frac{1}{2}\right)\cos x - \frac{\sqrt{3}}{2}\sin x. \]

Step 3: Combine and Verify

Now we will substitute both simplified expressions into the given equation: \[ \sin(210^{\circ} + x) - \cos(120^{\circ} + x) = \left( -\frac{1}{2}\cos x - \frac{\sqrt{3}}{2}\sin x \right) - \left( -\frac{1}{2}\cos x - \frac{\sqrt{3}}{2}\sin x \right). \] By distributing the negative sign, we notice that both terms are exactly the same: \[ 0 = 0. \]Since both sides of the equation balance perfectly, the given equation is indeed an identity.

Key Concepts

Angle Addition Formulas

The angle addition formulas are fundamental tools in trigonometry used to simplify expressions involving sums of angles. These formulas allow us to express trigonometric functions of angle sums, such as \( \sin(A + B) \) and \( \cos(A + B) \), in terms of products of sine and cosine functions of the individual angles. For the sine addition formula, we use:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

Similarly, the cosine addition formula is:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

Understanding these formulas is crucial for solving trigonometric identities, as it allows for the transformation and simplification of complex equations into more manageable forms. With the help of these formulas, one can break down challenging problems into simpler, solvable steps.

Sine Function

The sine function is one of the primary trigonometric functions, often used to define the relationship between the angles and sides of a right triangle. The sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the context of angle addition, the sine function's values at specific angles, such as \( \sin 210^{\circ} \), are essential for applying the sine addition formula accurately. For example, \( \sin 210^{\circ} = -\frac{1}{2} \), which is derived from the unit circle and helps in simplifying expressions involving angle sums. Understanding the specific values of the sine function for standard angles allows for correctly substituting in formulas and solving equations.

Cosine Function

The cosine function is another crucial trigonometric function, depicting the adjacent side of a right triangle in relation to the hypotenuse:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

In solving trigonometric identities, especially ones involving angle addition formulas, it's important to know common values such as \( \cos 120^{\circ} = -\frac{1}{2} \). This value is key for calculations that require breaking down the cosine of a sum like \( \cos(120^{\circ} + x) \). By referencing these known values from the unit circle, one can apply the cosine addition formula effectively, hence facilitating simplification and verification of trigonometric expressions.

Verification of Identities

Verifying trigonometric identities involves proving that two different looking expressions are indeed equivalent. The objective is to use known identities and formulas to manipulate one side of the equation to look exactly like the other side. In the given exercise, the usage of angle addition formulas allowed for the breakdown and consequent simplification of both the sine and cosine terms. The process of distributing and rearranging verified that the expressions were indeed equal, satisfying the identity \( \sin(210^{\circ} + x) - \cos(120^{\circ} + x) = 0 \).
This process often includes:

• Identifying which identities or formulas are applicable to the problem.
• Simplifying each side independently and systematically.
• Ensuring all algebraic manipulations are correct.

Remember, the essence of verifying identities is not just about finding the solution but understanding why and how the solution works, reinforcing trigonometric concepts fundamentally.