Problem 10 - Algebra 2

Question

Verify that each of the following is an identity. \(\sin \left(\theta+30^{\circ}\right)+\cos \left(\theta+60^{\circ}\right)=\cos \theta\)

Step-by-Step Solution

Answer

The given equation is an identity because both sides simplify to \( \cos \theta \).

Step 1: Use Angle Sum Formulas

We will use the angle sum formulas for trigonometric functions. The formula for sine is: \( \sin (A + B) = \sin A \cos B + \cos A \sin B \) and for cosine it is: \( \cos (A + B) = \cos A \cos B - \sin A \sin B \).

Step 2: Expand \( \sin(\theta + 30^{\circ}) \)

Using the formula for \( \sin(A+B) \) with \( A = \theta \) and \( B = 30^{\circ} \), we get: \[\sin(\theta + 30^{\circ}) = \sin \theta \cos 30^{\circ} + \cos \theta \sin 30^{\circ}.\]Since \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \), substituting these values in gives: \[\sin(\theta + 30^{\circ}) = \sin \theta \cdot \frac{\sqrt{3}}{2} + \cos \theta \cdot \frac{1}{2}.\]

Step 3: Expand \( \cos(\theta + 60^{\circ}) \)

Similarly, using the formula for \( \cos(A+B) \) with \( A = \theta \) and \( B = 60^{\circ} \), we get: \[\cos(\theta + 60^{\circ}) = \cos \theta \cos 60^{\circ} - \sin \theta \sin 60^{\circ}.\]Since \( \cos 60^{\circ} = \frac{1}{2} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), substituting these values in gives: \[\cos(\theta + 60^{\circ}) = \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2}.\]

Step 4: Combine Expansions

Now we combine \( \sin(\theta + 30^{\circ}) \) and \( \cos(\theta + 60^{\circ}) \):\[\sin(\theta + 30^{\circ}) + \cos(\theta + 60^{\circ}) = \left( \sin \theta \cdot \frac{\sqrt{3}}{2} + \cos \theta \cdot \frac{1}{2} \right) + \left( \cos \theta \cdot \frac{1}{2} - \sin \theta \cdot \frac{\sqrt{3}}{2} \right).\]Simplifying the expression, the terms involving \( \sin\theta \) and \( \cos\theta \) cancel out, resulting in:\[\cos \theta.\]

Step 5: Conclude the Identity Verification

Since after simplification, we are left with \( \cos \theta \), which matches the right-hand side of the given identity, the given equation is indeed an identity:\[\sin(\theta + 30^{\circ}) + \cos(\theta + 60^{\circ}) = \cos \theta.\]

Key Concepts

Angle Sum Formulas

The angle sum formulas are fundamental tools in trigonometry used when working with the sum or difference of two angles. These formulas allow us to express trigonometric functions of a compound angle in terms of the functions of individual angles. This can simplify a problem significantly.

• The formula for sine sum is: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
• The formula for cosine sum is: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).

In the exercise, these formulas enable us to break down complex angles like \( \theta + 30^{\circ} \) and \( \theta + 60^{\circ} \) into more manageable terms. This step is crucial for verifying trigonometric identities, where simplifying either side of an equation to show they're equal is often desired.

Sine and Cosine Functions

The sine and cosine functions are the backbone of trigonometry. They are based on right-angled triangles and vital in describing periodic phenomena.

• Sine Function: Describes the ratio of the length of the opposite side to the hypotenuse in a right-angle triangle.
• Cosine Function: Describes the ratio of the length of the adjacent side to the hypotenuse.

The problem requires these functions' values at specific angles, such as 30° and 60°. Knowing these values:

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)

helps us apply the angle sum formulas and verify identities. Understanding sine and cosine's properties, like periodicity and symmetry, can further aid in making sense of how identities unfold.

Identity Verification

Identity verification in trigonometry involves proving that two expressions are equivalent for all values of the variable. This process is crucial for understanding the intrinsic relations between trigonometric functions.

• First, apply the angle sum formulas to decompose the angles into their sine and cosine components. This was done in Step 2 and 3 of the solution, allowing simplification.
• Next, simplify the expressions by combining like terms, as shown in Step 4, which often cancels terms, leading to easy comparison to the desired identity.
• Finally, ensure both sides of the original equation match, confirming the identity, as concluded in Step 5.

Identity verification builds confidence in the reliability of trigonometric functions and strengthens problem-solving skills. By practicing with these identities, understanding abstracts into usability in advanced mathematical concepts.