Problem 15 - Algebra and Trigonometry

Question

Find the exact value of the expression, if it is defined. \(\tan \left(\tan ^{-1} 5\right)\)

Step-by-Step Solution

Answer

The exact value of the expression is 5.

Step 1: Understand the Problem

We are asked to find the exact value of the expression \( \tan(\tan^{-1}(5)) \). This is an application of the inverse trigonometric functions.

Step 2: Analyze \( \tan^{-1}(5) \)

The expression \( \tan^{-1}(5) \) represents the angle \( \theta \) such that \( \tan(\theta) = 5 \). This means \( \tan^{-1}(5) \) essentially gives us an angle whose tangent is 5.

Step 3: Apply Inverse Property

The mathematical property of the tangent function and its inverse states that if \( y = \tan^{-1}(x) \), then \( \tan(y) = x \). In our case, if \( \theta = \tan^{-1}(5) \), then it follows that \( \tan(\theta) = 5 \).

Step 4: Substitute Back

Since \( \theta = \tan^{-1}(5) \) and \( \tan(\theta) = 5 \), substituting back, we get \( \tan(\tan^{-1}(5)) = 5 \). So the exact value of \( \tan(\tan^{-1}(5)) \) is 5.

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions, along with sine and cosine. It is often abbreviated as "tan." In a right-angled triangle, the tangent of an angle is the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. This can be expressed as:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]The tangent function can take on any real value, which is especially important when dealing with its inverse function, since all real numbers are possible outputs for tangent. When graphing, the function repeats its values in a periodic interval of \(\pi\) (180 degrees). In real numbers, the tangent function is undefined at odd multiples of \(\frac{\pi}{2}\).

Some characteristics of the tangent function include:

• Periodic with period \(\pi\)
• Continuous except for points where it is undefined
• Range of all real numbers

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all allowed values of the occurring variables. These identities are fundamental in simplifying expressions and solving trigonometric equations. They can also be instrumental when dealing with symmetrical properties of angles and periodic functions.

Some important trigonometric identities include:

• Pythagorean Identities: \(\sin^2(x) + \cos^2(x) = 1\)
• Angle Sum and Difference Identities: \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)
• Double Angle Identities: \(\sin(2x) = 2 \sin(x) \cos(x)\)

The inverse properties, such as \(\tan(\tan^{-1}(x)) = x\), are also crucial. These identities showcase the inverse relationship between functions like tangent and arctangent, simplifying the evaluation of expressions like the one in our exercise.

Angle Measurement

Angle measurement is fundamental in trigonometry, serving as the standard for expressing the extent to which two lines diverge from a common point (vertex). Angles can be measured in degrees or radians.

Degrees are a historically common unit, where a full circle is divided into 360 equal parts, known as degrees. Radians, on the other hand, are a unit based on the radius of the circle, where the angle is defined as the length of the arc divided by the radius. A full circle corresponds to \(2\pi\) radians, which is equivalent to 360 degrees. The conversion between the two is given by:\[1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\]For practical applications, radians are often preferred in higher mathematics and calculus because they facilitate simpler derivations and integrations. Angles in trigonometric functions typically are in radians within mathematical analysis, ensuring natural periodicity and simplification.