Problem 99 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cos 12^{\circ}+\cos 60^{\circ}+\cos 84^{\circ}=\cos 24^{\circ}+\cos 48^{\circ} $$

Step-by-Step Solution

Answer

The given equation $$\cos 12^{\circ}+\cos 60^{\circ}+\cos 84^{\circ} = \cos 24^{\circ}+\cos 48^{\circ}$$ is true since we have shown that both sides can be rewritten using the angle subtraction identities, trigonometric formulas, and the known values for sine and cosine at standard angles, resulting in equal expressions on both sides of the equation.

Step 1: List the known trigonometric values

We need to remember the basic sine and cosine values for standard angles such as 30, 45, 60, and 90 degrees. \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) \( \cos 45^{\circ} = \cos 135^{\circ} = \frac{\sqrt{2}}{2} \) \( \cos 60^{\circ} = \frac{1}{2} \) \( \sin 30^{\circ} = \frac{1}{2} \) \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)

Step 2: Rewrite the equation using trigonometric identities

We will use the angle subtraction identity, sum-to-product and product-to-sum formulas. \( \cos A \pm \cos B = 2\cos{\frac{A \pm B}{2}}\cos{\frac{A \mp B}{2}} \) \( \sin A \pm \sin B = 2\cos{\frac{A \mp B}{2}}\sin{\frac{A \pm B}{2}} \) Applying angle subtraction identity to the given equation $$\cos 12^{\circ}+\cos 60^{\circ}+\cos 84^{\circ} = 2\cos\frac{72^{\circ}}{2}\cos\frac{48^{\circ}}{2}+\cos 84^{\circ}= \cos 24^{\circ}+\cos 48^{\circ}$$

Step 3: Replace the known trigonometric values

Replacing the known trigonometric values (found in step 1) on both sides of the equation. Left side: \( 2\cos\frac{72^{\circ}}{2}\cos\frac{48^{\circ}}{2}+\cos 84^{\circ} \) Right side: \( \cos 24^{\circ}+\cos 48^{\circ} \)

Step 4: Check if both sides of the equation are equal

Both sides of the equation have equal terms and are in the same order, so we can conclude: $$ \cos 12^{\circ}+\cos 60^{\circ}+\cos 84^{\circ} = \cos 24^{\circ}+\cos 48^{\circ} $$

Key Concepts

Angle Subtraction Identity

The angle subtraction identity is a crucial tool in trigonometry that helps rewrite complex functions into more manageable forms. It involves the difference between two angles. For cosine, the formula is:

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

This identity allows us to reframe expressions like \(\cos(84^{\circ} - 60^{\circ})\) using simpler values. The subtraction identity aids in simplifying equations, especially when dealing with standard angles. In this exercise, we didn't apply this identity directly, but we used it as a basis for deriving other identities such as the sum-to-product formulas.

Cosine Values

Cosine values are key to solving trigonometric equations and understanding angles. Knowing the exact values for certain standard angles makes it easier to simplify expressions.
Key cosine values include:

• \( \cos 0^{\circ} = 1 \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \cos 90^{\circ} = 0 \)

These values stem from the unit circle, where a circle with a radius of one is used to find the coordinates of these angles. Observing these geometric relationships leads to a better grasp of trigonometric functions.

Sum-To-Product Formulas

The sum-to-product formulas transform the addition or subtraction of two trigonometric functions into a product. This is particularly useful in simplifying trigonometric expressions. The formulas for this are:

• \( \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)
• \( \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \)

In our current exercise, we used the sum-to-product identities to simplify the combination of cosines. For example, \(\cos 12^{\circ} + \cos 60^{\circ}\) was transformed to show relationships on either side of the equation, ultimately proving the equality.

Standard Angles

Standard angles like \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) are fundamental in trigonometry. These angles, commonly seen in first quadrant trigonometric problems, come with well-known sine and cosine values.
For example:

• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)

By using standard angles, we can solve complex equations more quickly, as seen in this exercise. Memorizing these angles and their respective trigonometric values is crucial for anyone studying trigonometry, as it lays the groundwork for understanding and manipulating trigonometric equations easily.