Question

Verify $$ \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B} $$ with \(A=65^{\circ}\) and \(B=30^{\circ}\).

Short answer

Question: Verify the formula \(\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\) using \(A = 65^{\circ}\) and \(B = 30^{\circ}\). Answer: The formula \(\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\) has been verified using \(A = 65^{\circ}\) and \(B = 30^{\circ}\) as both the LHS and RHS values are approximately equal to \(0.7002\).

Step-by-step solution

Calculate LHS (tan(A-B))

Using the given values of \(A = 65^{\circ}\) and \(B = 30^{\circ}\), find \(\tan(A-B)\): $$ \tan(65^{\circ} - 30^{\circ}) = \tan(35^{\circ}) $$

Find the value of tan(35°)

Using a calculator or trigonometric table, find the value of \(\tan(35^{\circ})\): $$ \tan(35^{\circ}) \approx 0.7002 $$ So LHS = \(\tan(35^{\circ}) \approx 0.7002\)

Calculate RHS (tan A - tan B)/(1 + tan A * tan B)

Using the given values of \(A = 65^{\circ}\) and \(B = 30^{\circ}\), find \(\frac{\tan A-\tan B}{1+\tan A \tan B}\): $$ \frac{\tan 65^{\circ} - \tan 30^{\circ}}{1 + \tan 65^{\circ} \tan 30^{\circ}} $$

Find the values of tan(65°) and tan(30°)

Using a calculator or trigonometric table, find the values of \(\tan(65^{\circ})\) and \(\tan(30^{\circ})\): $$ \tan(65^{\circ}) \approx 2.1445 $$ $$ \tan(30^{\circ}) \approx 0.5774 $$

Substitute tan(65°) and tan(30°) in RHS formula

Substitute the values of \(\tan(65^{\circ})\) and \(\tan(30^{\circ})\) in the RHS formula: $$ \frac{2.1445 - 0.5774}{1 + 2.1445 \cdot 0.5774} \approx \frac{1.5671}{2.2376} $$

Evaluate RHS

Evaluate the expression obtained in step 5: $$ \frac{1.5671}{2.2376} \approx 0.7002 $$ So RHS = \(0.7002\)

Compare LHS and RHS

Compare the values of LHS and RHS that we calculated in steps 2 and 6: LHS = \(0.7002\) RHS = \(0.7002\) Since LHS = RHS, the formula has been verified with \(A = 65^{\circ}\) and \(B = 30^{\circ}\).

Key concepts

Understanding the Tangent Function

The tangent function, often abbreviated as 'tan', is a fundamental concept in trigonometry. It is one of the three primary trigonometric ratios, alongside sine and cosine. Tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. In simpler terms, for a given angle \theta, we can write the definition as:
\[\begin{equation}\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\end{equation}\]
Moreover, this function is periodic with a period of \(\pi\) radians (or 180 degrees), meaning the values of the tan function repeat every 180 degrees. This is the reason why \(\tan(A - B)\) can be evaluated for any angles A and B through their difference.

Using 'tan' in Real-World Problems

In real scenarios, the tan function is used to determine steepness or the slope of an incline, which is crucial in fields such as engineering and architecture. In our example exercise of verifying trigonometric identities, understanding how to compute \(\tan\) values for specific angles plays a critical role in moving forward to the more complex topic of trigonometric formulas.

Trigonometric Formulas

Trigonometric formulas are the backbone of trigonometry. They allow students and mathematicians to relate the angles and sides of triangles, navigate problems involving periodic functions, and even solve complex equations involving trigonometric terms.
The formula in question for our exercise, \(\tan (A-B) = \frac{\tan A-\tan B}{1+\tan A \tan B}\), represents one of these vital identities. It is a part of the angle subtraction properties and it succinctly captures how the tangent values of two different angles can be related when these angles are subtracted. This particular formula is useful in applications such as calculus, where function arguments are frequently manipulated for analysis, integration, or differentiation.

Application of Trigonometric Formulas

These formulas are not only academic exercises; they have substantial real-life implications. For instance, in physics, they are used to resolve forces, in navigation to set course directions, and in computer graphics for rotations and scaling operations. By mastering them, students can better understand the mathematical structure of oscillations, waves, and other phenomena that repeat over time or space.

Angle Subtraction Properties

Angle subtraction properties are crucial when working with trigonometric functions. They give us valuable tools for simplifying expressions where the angle is, as the name suggests, a difference between two other angles. Specifically, for the tangent function, the angle subtraction property comes in the form of the previously mentioned formula: \(\tan (A-B)\).
This identity is derived from the sine and cosine subtraction formulas, reflecting the interconnectedness of trigonometric identities. We can formally state the property as:
\[\begin{equation}\tan(A-B) = \frac{\sin(A-B)}{\cos(A-B)} = \frac{\sin A \cos B - \cos A \sin B}{\cos A \cos B + \sin A \sin B} = \frac{\tan A - \tan B}{1 + \tan A \tan B}\end{equation}\]
With such properties, the simplification of trigonometric expressions becomes feasible, revealing more straightforward paths to solutions.

Importance in Various Fields

The angle subtraction properties are not just theoretical; in fields like engineering and physics, they can be used to analyze wave functions, sound patterns, and much more. For students, getting comfortable with these concepts paves the way for advancements in not only trigonometry but also calculus and other higher mathematical studies.