Problem 31 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \tan \left(\sin ^{-1} \frac{12}{13}\right) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{12}{5} \).

Step 1: Understand the Problem

We need to find the value of \( \tan \left( \sin^{-1} \frac{12}{13} \right) \). Essentially, we're looking for the tangent of an angle whose sine is \( \frac{12}{13} \).

Step 2: Use the Right Triangle

Since we are dealing with \( \sin^{-1} \frac{12}{13} \), think of a right triangle where the angle \( \theta \) has a sine of \( \frac{12}{13} \). This means the opposite side is \( 12 \) and the hypotenuse is \( 13 \). We need to find the adjacent side to determine the tangent.

Step 3: Apply the Pythagorean Theorem

Using the Pythagorean Theorem \( a^2 + b^2 = c^2 \), where \( a \) is the adjacent side, \( b \) is \( 12 \), and \( c \) is \( 13 \). We find the adjacent side:\[a^2 + 12^2 = 13^2\]\[a^2 + 144 = 169\]\[a^2 = 25\]\[a = 5\]

Step 4: Compute the Tangent

The tangent of the angle \( \theta \) is the ratio of the opposite side to the adjacent side. Hence,\[\tan( \theta ) = \frac{ \text{opposite} }{ \text{adjacent} } = \frac{12}{5}\]

Step 5: Conclude with the Exact Value

The exact value of \( \tan( \sin^{-1} \frac{12}{13} ) \) is \( \frac{12}{5} \).

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. In simple terms, the tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side.
This can be mathematically expressed as:

• \[\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}.\]

The tangent function is extremely helpful when dealing with problems involving right triangles, as it allows you to relate two sides of the triangle using an angle. This trigonometric function is also periodic and repeats every \(180^\circ\) (or \(\pi\) radians), which means it adopts the same values at certain rotations.
The tangent function is an odd and non-linear function, directly reflecting its nature to reflect over the origin

• its graph passes through the origin and has vertical asymptotes (undefined values) where the function approaches infinity.

Understanding the tangent function is essential in trigonometry and plays a critical role in solving various geometric problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, used primarily in right-angled triangles. This theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In mathematical terms, if \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse, the theorem can be written as:

• \[a^2 + b^2 = c^2\]

The Pythagorean Theorem is not just a tool for calculating side lengths; it also establishes a foundation for trigonometric concepts and proofs. For the given exercise, knowing two sides of a right triangle allowed us to determine the missing side using this theorem, thus enabling us to compute the tangent function accurately.
Additionally, the Pythagorean Theorem can be generalized beyond geometric applications, being critical in fields like calculus, physics, and engineering.

Right Triangle

A right triangle is a special type of triangle that includes a right angle (\(90^\circ\)). It consists of three sides:

• two legs, which form the right angle, and
• the hypotenuse, which is the side opposite the right angle.

This simple geometric shape is foundational in trigonometry. Each angle in a right triangle is connected to the triangle’s sides through trigonometric ratios, such as sine, cosine, and tangent.
In our exercise, the right triangle facilitated our understanding through the sine of the angle. Knowing that \(\sin(\theta) = \frac{12}{13}\), we set one of the legs to \(12\) and the hypotenuse to \(13\). This helped us visualize the problem and use the Pythagorean Theorem to find the missing leg. Typically in such triangles, you can assign the known lengths to find other relationships or solve problems involving any missing angle or side.
Right triangles represent a crucial element in many areas of mathematics and are often a stepping stone to more advanced geometric and trigonometric concepts.