Problem 27 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \frac{\tan 10^{8}+\tan 35^{\circ}}{1-\tan 10^{\circ} \tan 35^{\circ}} $$

Step-by-Step Solution

Answer

The exact value of the given expression is 1.

Step 1: Recognize the identity

Firstly, recognition of the tan sum identity in the given expression is pivotal. The expression \( \frac{\tan 10^{8}+\tan 35^{\circ}}{1-\tan 10^{\circ} \tan 35^{\circ}} \) is similar to the \(\tan(A + B)\) identity.

Step 2: Rewrite the expression

Using the tan sum identity, \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \), rewrite the given expression in the form of \(\tan(A + B)\). Here, angle A is \( 10^{\circ}\) and angle B is \( 35^{\circ}\). Hence, the expression can be rewritten as \(\tan(10^{\circ} + 35^{\circ})\).

Step 3: Simplify the expression

Simplify \(\tan(10^{\circ} + 35^{\circ})\) to \(\tan 45^{\circ}\).

Step 4: Find the exact value

The final step is to find the exact value of \(\tan 45^{\circ}\). The tangent of 45 degrees is 1.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is key in solving a wide array of mathematical problems involving angles. These identities are equations that hold true for all values of the involved variables and relate the trigonometric functions to interior angles. In this exercise, we use the sum and difference identities. These particular identities help in simplifying expressions involving sums or differences of angles.
The tangent sum identity, specifically, is shown as \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).
Here are some benefits of understanding these identities:

• They help in simplifying complex trigonometric expressions.
• They allow for the transformation of expressions into more workable forms.
• They provide a foundation for solving both geometric and real-world problems involving periodic phenomena.

By mastering these trigonometric identities, you can tackle exercises involving angle calculations with more confidence.

Tangent Function

The tangent function is a key trigonometric function, often symbolized as \( \tan \). It describes the ratio of the opposite side to the adjacent side in a right triangle.
In our exercise, the tangent function forms the basis of the sum identity used in the solution. It assists in the transformation of the sum of angles into a single trigonometric expression.
Key properties of the tangent function include:

• Periodicity: The tangent function has a period of \(180^{\circ} \).
• Undefined at angles where the cosine is zero, such as \(90^{\circ} \), \(270^{\circ} \), etc.
• In the unit circle, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is another way to express it, linking it with sine and cosine functions.

These properties make the tangent function crucial in solving problems related to triangles and cycles. The solution utilizes these characteristics to resolve the expression into a calculable form.

Exact Trigonometric Values

Exact trigonometric values are predefined values for certain angles and are essential for precise calculations, avoiding rounding errors. These values commonly come from special angles such as \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), etc., which typically form the basis for many trigonometric calculations.
In this problem, the exact value of \( \tan 45^{\circ} \) is determined.
Important aspects to note include:

• The tangent of \(45^{\circ}\) is exactly 1 because, in a \(45^{\circ}\) triangle, the opposite and adjacent sides are equal, making the ratio 1:1.
• Using exact values helps prevent errors that might arise from using decimal approximations.
• These values are frequently used in exams and practice problems, so memorizing them can be quite beneficial.

Recognizing these exact values not only boosts computational speed but also sharpens your analytical skills in trigonometry.