Problem 92 - Algebra and Trigonometry

Question

Show that \(\cos 100^{\circ}-\cos 200^{\circ}=\sin 50^{\circ}\).

Step-by-Step Solution

Answer

The identity is true: \(\cos 100^{\circ} - \cos 200^{\circ} = \sin 50^{\circ}\).

Step 1: Apply Cosine Difference Identities

First, utilize the identity: \( \cos(A) - \cos(B) = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \).Set \( A = 100^{\circ} \) and \( B = 200^{\circ} \).Substitute these values into the identity:\[ \cos(100^{\circ}) - \cos(200^{\circ}) = -2 \sin\left(\frac{100^{\circ} + 200^{\circ}}{2}\right) \sin\left(\frac{100^{\circ} - 200^{\circ}}{2}\right) \]

Step 2: Simplify the Argument of Sines

Calculate \( \frac{100^{\circ} + 200^{\circ}}{2} \) and \( \frac{100^{\circ} - 200^{\circ}}{2} \).\[ \frac{100^{\circ} + 200^{\circ}}{2} = \frac{300^{\circ}}{2} = 150^{\circ} \]\[ \frac{100^{\circ} - 200^{\circ}}{2} = \frac{-100^{\circ}}{2} = -50^{\circ} \]

Step 3: Apply the Simplified Arguments

Now substitute \(150^{\circ}\) and \(-50^{\circ}\) into the cosine difference identity:\[ \cos(100^{\circ}) - \cos(200^{\circ}) = -2 \sin(150^{\circ}) \sin(-50^{\circ}) \]

Step 4: Use Sin Function Properties

Recall that \( \sin(-x) = -\sin(x) \), and substitute the property:\[ -2 \sin(150^{\circ}) \sin(-50^{\circ}) = -2 \sin(150^{\circ})(-\sin(50^{\circ})) = 2 \sin(150^{\circ}) \sin(50^{\circ}) \]

Step 5: Calculate the Known Value

Recall that \( \sin(150^{\circ}) = \sin(180^{\circ} - 30^{\circ}) = \sin(30^{\circ}) = \frac{1}{2} \).Substitute this in:\[ 2 \left(\frac{1}{2}\right) \sin(50^{\circ}) = \sin(50^{\circ}) \]

Step 6: Conclude the Solution

We have shown that:\[ \cos(100^{\circ}) - \cos(200^{\circ}) = \sin(50^{\circ}) \]Therefore, the identity holds true.

Key Concepts

Understanding Cosine Difference Identity

The cosine difference identity is an essential trigonometric formula that can simplify expressions involving the difference of two cosine functions. It is given by:

• \( \cos(A) - \cos(B) = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \)
• Here, \( A \) and \( B \) are angles, and the formula transforms the difference of cosines into a product of sines.

This transformation is particularly useful when you have to prove trigonometrical identities or simplify expressions. By converting cosine differences into a manageable sine form, complex problems can often be simplified significantly.
To apply this identity effectively, it's essential to substitute the correct angle values and carefully follow the arithmetic to ensure accurate simplification. As in our original exercise, taking \( A = 100^{\circ} \) and \( B = 200^{\circ} \), the transformation leads directly to the ability to substitute known sine values and conclude the solution. This identity is a tool that streamlines angle calculations, making it easier to deal with seemingly complicated expressions.

Exploring Sine Function Properties

The sine function has several important properties that make it powerful in trigonometric calculations. One of the key properties is the function's symmetry:

• \( \sin(-x) = -\sin(x) \) demonstrates that the sine function is an odd function, mirroring its point across the origin of its graph.
• This property is crucial when handling angles with negative measures, as it allows us to change the sign conveniently.

Another related property is the sine of complementary angles:

• \( \sin(180^{\circ} - x) = \sin(x) \), which reflects the concept that sine values repeat in a symmetrical pattern about \(90^{\circ}\) and \(180^{\circ}\).

These properties were used directly in our original exercise. For example, knowing that \( \sin(150^{\circ}) = \sin(30^{\circ}) = \frac{1}{2} \) simplifies the calculation of our final proof. Such properties help in identifying equivalent sine values efficiently and revealing paths to solve trigonometric problems more effortlessly.

Calculating Angles Effectively

Angle calculations in trigonometry often revolve around figuring out the correct measures to use within identities and simplifying expressions. When calculating angles:

• Start by accurately determining any sum or difference as needed for identities like the cosine difference identity.
• For example, finding \( \frac{100^{\circ} + 200^{\circ}}{2} = 150^{\circ} \) and \( \frac{100^{\circ} - 200^{\circ}}{2} = -50^{\circ} \) allows for meaningful substitution and application of trigonometric identities.
• It's important to simplify these angle measures to key reference angles that commonly occur in the unit circle, such as \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), etc.

By translating complex or unusual angles into these fundamental measures, you gain access to known trigonometric values. These calculated values readily fit into identities or are replaced by known ratios efficiently, expediting calculations and helping to ensure accuracy in finding solutions.