Problem 203 - Precalculus

Question

For the following exercises, find the exact value without the aid of a calculator. $$ \cot \left(\sin ^{-1}\left(\frac{3}{5}\right)\right) $$

Step-by-Step Solution

Answer

\( \frac{4}{3} \)

Step 1: Understand the Expression

The expression \[ \cot \left(\sin^{-1}\left(\frac{3}{5}\right)\right) \] requires finding the cotangent of the angle whose sine is \( \frac{3}{5} \). This angle is denoted by \( \theta \), where \( \sin \theta = \frac{3}{5} \).

Step 2: Sketch a Right Triangle

Consider a right triangle where \( \theta \) is one of the angles. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. Therefore, let the opposite side be 3 and the hypotenuse be 5.

Step 3: Apply the Pythagorean Theorem

To find the length of the adjacent side, use the Pythagorean theorem: \[ hypotenuse^2 = opposite^2 + adjacent^2 \] \[ 5^2 = 3^2 + adjacent^2 \] \[ 25 = 9 + adjacent^2 \] \[ adjacent^2 = 16 \] \[ adjacent = 4 \] Now, the lengths of the sides are: opposite = 3, adjacent = 4, hypotenuse = 5.

Step 4: Calculate the Cotangent

The cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side: \[ \cot \theta = \frac{adjacent}{opposite} \] \[ \cot \theta = \frac{4}{3} \]

Step 5: Write the Final Result

Thus, the exact value of \[ \cot \left(\sin^{-1}\left(\frac{3}{5}\right)\right) \] is \( \frac{4}{3} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help you determine angles when you know the trigonometric ratio. In other words, if you know that the sine of an angle is a specific value, say \( \frac{3}{5} \), the inverse sine function, written as \( \sin^{-1} \) or arcsin, helps you find the angle with that sine value.
They are crucial in many fields, including engineering and physics, as they allow you to translate real-world measurements into angles.
In solving trigonometric identities, using these inverse functions effectively lets you express angles in terms of familiar ratios. They convert numeric values back into angles, which can then be used to find other trigonometric identities like cotangent, tangent, etc.

• \( \sin^{-1} \) returns an angle whose sine is the value within its parentheses.
• These functions are restricted to specific intervals to maintain their properties as functions. For example, \( \sin^{-1} x \) is only defined for \( -1 \leq x \leq 1 \).

Right Triangle Trigonometry

Right triangle trigonometry is the study of the relationships between the angles and sides of right triangles. This field lays the foundation for solving problems involving angles and lengths, especially when relating to trigonometric functions.
Each function (sine, cosine, tangent, etc.) is a ratio of two sides of a right triangle:

• Sine (\( \sin \)): Ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \)): Ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)): Ratio of the opposite side to the adjacent side.
• Cotangent (\( \cot \)): Ratio of the adjacent side to the opposite side.

These relationships are essential when solving problems like the one presented in the original exercise where knowing one side allows you to find the others. The celebrated Pythagorean theorem, expressed as \( a^2 + b^2 = c^2 \), allows for the calculation of a missing side length when two sides are known.

Cotangent Function

The cotangent function, denoted \( \cot \), is one of the six fundamental trigonometric functions. It relates the angles of a triangle to the ratios of two of its sides. For an angle \( \theta \) in a right triangle, \( \cot \theta \) is defined as the ratio of the adjacent side to the opposite side.
Mathematically, this is written as:
\[\cot \theta = \frac{\text{adjacent}}{\text{opposite}}\]

• It is the reciprocal of the tangent function: \( \cot \theta = \frac{1}{\tan \theta} \).
• This function is useful for finding unknown side lengths from known ratios or for solving equations that involve right triangles.

In the context of the problem, since the triangle's adjacent side is 4 and the opposite side is 3, the cotangent of \( \theta \) (where \( \sin \theta = \frac{3}{5} \)) becomes \( \frac{4}{3} \). Understanding \( \cot \) allows you to navigate trigonometric problems by leveraging your knowledge of ratios and angles, building a better grasp of how these functions interrelate within a right triangle context.