Problem 94 - Precalculus Mathematics for Calculus

Question

Show that \(\cos 87^{\circ}+\cos 33^{\circ}=\sin 63^{\circ}\)

Step-by-Step Solution

Answer

\(\cos 87^{\circ} + \cos 33^{\circ} = \sin 63^{\circ}\) is true using trigonometric identities.

Step 1: Understanding the Objective

The goal is to prove that the expression \(\cos 87^{\circ} + \cos 33^{\circ}\) is equal to \(\sin 63^{\circ}\). This requires us to use trigonometric identities.

Step 2: Use Complementary Angles

Recall that \(\cos(90^{\circ} - x) = \sin(x)\). So, \(\cos 87^{\circ} = \sin(90^{\circ} - 87^{\circ}) = \sin 3^{\circ}\).

Step 3: Express Cosines as Sines

Using the identity from Step 2: \(\cos 33^{\circ} = \sin(90^{\circ} - 33^{\circ}) = \sin 57^{\circ}\).

Step 4: Use Sine Sum Formula

The sine sum identity is \(\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)\). Apply this with \(A = 57^{\circ}\) and \(B = 3^{\circ}\):\[\sin 3^{\circ} + \sin 57^{\circ} = 2 \sin\left(\frac{60^{\circ}}{2}\right) \cos\left(\frac{54^{\circ}}{2}\right)\]

Step 5: Simplify the Expression

Calculate the expressions: \[\sin(30^{\circ}) = \frac{1}{2}\quad \text{and} \quad \cos(27^{\circ})\] Thus, the identity becomes:\[\sin 3^{\circ} + \sin 57^{\circ} = 2 \times \frac{1}{2} \times \cos 27^{\circ} = \cos 27^{\circ}\]

Step 6: More Angle Conversion

Express \(\cos 27^{\circ}\) in terms of sine for further simplification if necessary. However, \(\cos 27^{\circ} = \sin(90^{\circ} - 27^{\circ}) = \sin 63^{\circ}\), which completes the proof.

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This concept is especially useful in trigonometry because of its relation to sine and cosine functions. The key identity here is \( \cos(90^{\circ} - x) = \sin(x) \). This means the cosine of an angle can be rewritten as the sine of its complementary angle. This identity helps in simplifying trigonometric expressions and solving equations.

• Example: For \( 87^{\circ} \), the complementary angle is \( 3^{\circ} \).
• This lets us transform \( \cos 87^{\circ} \) into \( \sin 3^{\circ} \).
• Similarly, \( \cos 33^{\circ} \) becomes \( \sin 57^{\circ} \), since \( 90^{\circ} - 33^{\circ} = 57^{\circ} \).

This identity bridges cosine and sine, helping to simplify and solve trigonometric equations by converting between these two functions.

Sine Sum Formula

The sine sum formula is a trigonometric identity that helps in adding two sine expressions. It is given by \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \). This formula comes in handy when working with problems that involve summing sine terms. It simplifies the process by transforming two distinct sine terms into a product, which can often be easier to manage.
Here’s how it works:

• For \( \sin 3^{\circ} + \sin 57^{\circ} \), apply this formula with \( A = 57^{\circ} \) and \( B = 3^{\circ} \).
• First, find the average of the angles: \( \frac{57^{\circ} + 3^{\circ}}{2} = 30^{\circ} \).
• Then, calculate the difference: \( \frac{57^{\circ} - 3^{\circ}}{2} = 27^{\circ} \).

This identity is useful in simplifying sums of sines to products of sines and cosines and can lead to revealing known angles or identities.

Angle Conversion

Angle conversion is a method used to switch between different representations of angles, often to simplify expressions or identify equivalent functions. A common technique is to convert between sine and cosine using complementary angles.

• For example, \( \cos 27^{\circ} \) can be rewritten using its complementary angle: \( \sin(90^{\circ} - 27^{\circ}) = \sin 63^{\circ} \).
• This conversion was used in our problem to show that \( \cos 27^{\circ} \) is actually \( \sin 63^{\circ} \), completing the proof that \( \cos 87^{\circ} + \cos 33^{\circ} = \sin 63^{\circ} \).

Angle conversion is handy for making connections between different trigonometric functions and simplifying expressions, providing greater flexibility in solving problems. It often aids in understanding relationships between various angles and their trigonometric identities.