Problem 138 - Precalculus

Question

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. $$ \sin \left(\cos ^{-1}\left(\frac{3}{5}\right)\right) $$

Step-by-Step Solution

Answer

\(\frac{4}{5}\)

Step 1: Understand the Given Expression

We are asked to find the exact value of \( \sin \left( \cos^{-1} \left( \frac{3}{5} \right) \right) \). Here, \( \cos^{-1} \left( \frac{3}{5} \right) \) means we are looking for an angle \( \theta \) such that \( \cos \theta = \frac{3}{5} \).

Step 2: Analyze the Problem within a Right Triangle

Since \( \cos \theta = \frac{3}{5} \), this means that if we consider a right triangle where \( \theta \) is an angle, the adjacent side to \( \theta \) is 3 and the hypotenuse is 5.

Step 3: Use the Pythagorean Theorem

We apply the Pythagorean Theorem to find the opposite side: \[ x^2 + 3^2 = 5^2 \]\[ x^2 + 9 = 25 \]\[ x^2 = 16 \]\[ x = 4 \]Thus, the opposite side is 4.

Step 4: Compute the Sine of the Angle

The sine of the angle \( \theta \) is given by the ratio of the opposite side to the hypotenuse in a right triangle. Therefore:\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \]

Step 5: Express the Final Answer

The exact value of \( \sin \left( \cos^{-1} \left( \frac{3}{5} \right) \right) \) is \( \frac{4}{5} \).

Key Concepts

Right Triangle

A right triangle is a triangle that has one angle measuring exactly 90 degrees. This type of triangle is fundamental in trigonometry because it provides a simple context for defining the basic trigonometric functions: sine, cosine, and tangent.
In a right triangle, there are three sides:

• Hypotenuse: The longest side, opposite the right angle.
• Adjacent Side: The side next to the angle of interest but not the hypotenuse.
• Opposite Side: The side opposite the angle of interest.

When examining a right triangle, such as in our problem, we often use it to establish relationships between the sides given a specific angle. Understanding these relationships is crucial to solving problems involving inverse trigonometric functions.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is:\[a^2 + b^2 = c^2\]where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse.
In the exercise, the theorem helps us find the length of the opposite side. Given that \( \cos \theta = \frac{3}{5} \), the adjacent side is 3, and the hypotenuse is 5. Applying the Pythagorean Theorem:

• \(3^2 + x^2 = 5^2\)
• \(9 + x^2 = 25\)
• \(x^2 = 16\)
• \(x = 4\)

This calculation confirms that the opposite side, used later to find the sine value, is 4.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the involved variables where both sides of the identity are defined. These identities help simplify complex trigonometric expressions.
For solving the original problem, we use the basic definition of sine, which is a primary trigonometric function in a right triangle:

• \(\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\)

Given \(\cos \theta = \frac{3}{5}\), the problem provides an angle \(\theta\) where the adjacent side is 3 and hypotenuse is 5. Using the Pythagorean Theorem, we've established the opposite side is 4. Thus, the sine of \(\theta\) can be calculated as:

• \(\sin \theta = \frac{4}{5}\)

This exact value can be found without using a calculator, demonstrating the power and utility of trigonometric identities in conjunction with geometric principles.