Problem 80 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Using trigonometric identities, \( \cos 87^{\circ} + \cos 33^{\circ} = \sin 63^{\circ} \).

Step 1: Understanding the Problem

We need to show that the expression \( \cos 87^{\circ} + \cos 33^{\circ} \) is equal to \( \sin 63^{\circ} \). To do so, we'll use trigonometric identities to transform the cosine terms to match the sine term.

Step 2: Using Complementary Angles

Recall that \( \sin(\theta) = \cos(90^{\circ} - \theta) \). So, \( \cos 87^{\circ} = \sin(90^{\circ} - 87^{\circ}) = \sin 3^{\circ} \). Also, \( \cos 33^{\circ} = \sin(90^{\circ} - 33^{\circ}) = \sin 57^{\circ} \).

Step 3: Simplifying the Equation

Substitute the simplified terms: \( \cos 87^{\circ} + \cos 33^{\circ} = \sin 3^{\circ} + \sin 57^{\circ} \). We want to show this is equal to \( \sin 63^{\circ} \).

Step 4: Using Sine Addition Formula

Apply the sum-to-product identities for sine: \( \sin A + \sin B = 2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right) \). Let \( A = 57^{\circ} \) and \( B = 3^{\circ} \), so: \( \sin 3^{\circ} + \sin 57^{\circ} = 2 \sin \left(\frac{3^{\circ} + 57^{\circ}}{2}\right) \cos \left(\frac{3^{\circ} - 57^{\circ}}{2}\right) \).

Step 5: Calculation

Calculate the expressions: \( \frac{3^{\circ} + 57^{\circ}}{2} = 30^{\circ} \) and \( \frac{3^{\circ} - 57^{\circ}}{2} = -27^{\circ} \). Therefore, \( \sin 3^{\circ} + \sin 57^{\circ} = 2 \sin 30^{\circ} \cos (-27^{\circ}) \).

Step 6: Trigonometric Value Simplification

Since \( \sin 30^{\circ} = \frac{1}{2} \) and \( \cos (-27^{\circ}) = \cos 27^{\circ} \) (because cosine is an even function), the expression becomes \( 2 \times \frac{1}{2} \times \cos 27^{\circ} = \cos 27^{\circ} \).

Step 7: Final Comparison

Notice that \( \cos 27^{\circ} \) is actually \( \sin(90^{\circ} - 27^{\circ}) = \sin 63^{\circ} \). Thus, \( \cos 87^{\circ} + \cos 33^{\circ} = \sin 63^{\circ} \).

Step 8: Conclusion

Through trigonometric identities and simplifications, we have shown that \( \cos 87^{\circ} + \cos 33^{\circ} = \sin 63^{\circ} \), confirming the equation.

Key Concepts

Cosine Addition

The cosine addition formula is a useful trigonometric identity that allows us to find the cosine of the sum of two angles. However, in this particular exercise, we are more concerned with breaking down each cosine term separately to transform them effectively.
In general, the cosine addition formula can be expressed as:

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

This identity is slightly different from our exercise, where we are looking at the sum of two separate cosine values rather than the cosine of a sum. Still, understanding this foundational rule helps us recognize patterns and identify applicable identities in various scenarios.
In our exercise, we're transforming individual cosine terms using complementary angles, rather than directly applying the cosine addition. It's a crucial step that sets the stage for using other identities to prove the given equation.

Sine Addition

The sine addition concept comes into play when dealing with the sum of sine values, much like we see in the given step-by-step solution. Although the formal sine addition identity handles the sine of angles added together, we can manipulate separate sine values by converting them into a different format.
The sine addition identity is traditionally used as:

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

For this exercise, we apply the understanding of how sine terms add together by using another helpful identity known as the sum-to-product identities. These identities allow us to convert sine sums into products, which can then be simplified more easily, as shown in the steps. Understanding this interaction between sine and cosine through such identities broadens our ability to manipulate and solve complex trigonometric expressions.

Sum-to-Product Identities

Sum-to-product identities are powerful tools in trigonometry that help transform sums into products, which typically are easier to handle. They are particularly helpful in our example as we break down both sine and cosine terms into their simpler forms.
Consider the identity for sine:

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

This identity is employed effectively in our exercise to express the sum of sine terms such as \(\sin 3^{\circ} + \sin 57^{\circ}\) into a more manageable equation. By converting our sum into a product, we allow easier simplification, eventually leading to the desired form. Simplification using this identity relies on calculating intermediate angles such as \(\frac{3^{\circ} + 57^{\circ}}{2} = 30^{\circ}\). This ultimately demonstrates the flexibility and power of sum-to-product identities in solving trigonometric problems.

Complementary Angles

Complementary angles are two angles that add up to 90 degrees. Understanding the concept of complementary angles is vital as it simplifies the relationship between sine and cosine functions.
An important property of complementary angles is:

• \(\sin(90^{\circ} - \theta) = \cos\theta\)
• \(\cos(90^{\circ} - \theta) = \sin\theta\)

In our exercise, we've transformed cosine values into sine values by recognizing each angle's complementary counterpart.
For instance, \(\cos 87^{\circ}\) becomes \(\sin 3^{\circ}\), and \(\cos 33^{\circ}\) translates to \(\sin 57^{\circ}\). This clever conversion allows us to manipulate and eventually prove the given trigonometric identity. Mastering complementary angles provides an effective way to navigate through similar problems by enabling smooth transitions between sine and cosine expressions.