Problem 71 - Precalculus: A Concise Course

Question

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7 .)\) $$ \sin (\arccos x) $$

Step-by-Step Solution

Answer

\(\sin (\arccos x) = \sqrt{1 - x^2}\

Step 1: Interpret the function

We start by interpreting \(\arccos x\) as an angle in a right triangle. By definition, the cosine of this angle would be \(x\). We can sketch a right triangle where the adjacent side has length \(x\) and the hypotenuse has length \(1\). This can be done because by definition, the cosine of an angle is \(x/1\), or simply \(x\).

Step 2: Find the opposite side

In a right triangle, if we know two sides, we can use the Pythagorean theorem to find the third side. Our triangle has a hypotenuse of 1, an adjacent side of \(x\), and we can solve for the opposite side, let’s call it \(y\), using the formula \(y^2 = 1^2 - x^2\). Taking the square root of both sides, we have \(y = \sqrt{1 - x^2}\). Note that we only take the positive root because, in our right triangle, the side lengths are positive.

Step 3: Write the sine of the angle

With all sides of the right triangle known, now we can express \(\sin(\arccos x)\). By definition, the sine of an angle is the ratio of the opposite side to the hypotenuse. In our case, for the angle \(\arccos x\), the opposite side is \(\sqrt{1 - x^2}\) and the hypotenuse is \(1\). So, our required solution \(\sin(\arccos x) = \frac{\sqrt{1 - x^2}}{1} = \sqrt{1 - x^2}\).

Key Concepts

Right Triangle Trigonometry

Right triangle trigonometry is based on the relationships between the sides and angles of right triangles. A right triangle has one angle measuring 90 degrees. The other two angles are complementary, adding up to 90 degrees. Each angle's trigonometric ratios are based on opposite, adjacent, and hypotenuse sides.

• **Opposite side**: The side opposite the angle of interest.
• **Adjacent side**: The side next to the angle of interest, excluding the hypotenuse.
• **Hypotenuse**: The longest side, opposite the right angle.

Several trigonometric functions arise from these relationships:

• **Sine (\( ext{sin} heta\))**: Ratio of the opposite side to the hypotenuse.
• **Cosine (\( ext{cos} heta\))**: Ratio of the adjacent side to the hypotenuse.
• **Tangent (\( ext{tan} heta\))**: Ratio of the opposite side to the adjacent side.

Understanding these ratios allows us to solve for unknown sides or angles in right triangles using trigonometric identities. In our problem, we use these concepts to transform the expression \(\sin(\arccos x)\) into its algebraic form.

Inverse Trigonometric Functions

Inverse trigonometric functions are about finding the angle when we know a trigonometric ratio. They reverse the process of the regular trigonometric functions. For example, \( ext{arccos} x\) yields an angle whose cosine is \(x\).
Inverse functions are represented by adding a prefix 'arc' to the usual trigonometric function. They provide the corresponding angle in radians or degrees:

• **Arccosine (\( ext{arccos} x\))**: Gives the angle \( heta \) such that \( ext{cos} \theta = x\).
• **Arcsine (\( ext{arcsin} x\))**: Gives the angle \( heta \) such that \( ext{sin} \theta = x\).
• **Arctangent (\( ext{arctan} x\))**: Gives the angle \( heta \) such that \( ext{tan} \theta = x\).

In our exercise, \(\text{arccos }x\) is used to determine an angle's cosine, enabling us to construct a right triangle and solve for other sides. This helps in finding alternate trigonometric ratios like sine for that angle.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry dealing with the side lengths of right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The theorem is written as:

• \[ c^2 = a^2 + b^2 \]

Where:

• \(c\) is the length of the hypotenuse,
• \(a\) and \(b\) are the lengths of the other two sides.

For our problem, using the Pythagorean theorem allows us to determine the unknown side of the triangle when the adjacent side is known to be \(x\) and the hypotenuse is \(1\). By rearranging, we find the opposite side:

• \[ y^2 = 1^2 - x^2 \]
• \[ y = \sqrt{1 - x^2} \]

This calculation provides the last piece of information needed to find the sine ratio for the angle \(\arccos x\), illustrating the power of blending trigonometric concepts with this classic theorem.