Question

Write the expression in algebraic form. $$ \sin \left(\cos ^{-1} x\right) $$

Short answer

The algebraic form of the expression \(\sin(\cos^{-1}x)\) is \(\sqrt{1 - x^2}\).

Step-by-step solution

Draw a Right Triangle and Label its Sides

Draw a right triangle and label one of the acute angles as \(\alpha\). Let the adjacent side length be \(a\), the opposite side length be \(b\), and the hypotenuse be \(c\).

Assign the Value of cosine to the Angle

Recall that\(\cos^{-1}x\) is the angle whose cosine is equal to \(x\). Now we assign the cosine of the angle \(\alpha\) as \(x\), so: \[ \cos \alpha = x \] Since cosine is defined as the ratio of the adjacent side to the hypotenuse, we can write the equation as follows: \[ \frac{a}{c} = x \] Now we can express one of the sides in terms of the others: \[ a = cx \]

Use the Pythagorean Theorem to Find the Other Side

Use the Pythagorean theorem for our right triangle, which states: \[ a^2 + b^2 = c^2 \] Substitute the expression for \(a\) from step 2 into this equation: \[ (cx)^2 + b^2 = c^2 \] Solve for \(b\): \[ b^2 = c^2 - c^2x^2 \] Now, take the square root of both sides: \[ b = \sqrt{c^2(1 - x^2)} \]

Determine the Expression for sine

Now we need to find \(\sin(\cos^{-1}x) = \sin \alpha\). Recall that sine is defined as the ratio of the opposite side to the hypotenuse: \[ \sin \alpha = \frac{b}{c} \] Then, substitute the expression for \(b\) from step 3: \[ \sin \alpha = \frac{\sqrt{c^2(1 - x^2)}}{c} \]

Simplify the Expression

Finally, simplify the expression: \[ \sin \alpha = \sqrt{1 - x^2} \] Now, our expression in algebraic form is: \[ \sin(\cos^{-1}x) = \sqrt{1 - x^2} \]

Key concepts

Sine and Cosine

Understanding the sine and cosine functions is essential when dealing with trigonometry, specifically in the context of right triangles. Sine and cosine are fundamental trigonometric ratios that compare the lengths of the sides of a right triangle.

Let's take a right triangle with an angle \( \theta \). The sine of \( \theta \) is the ratio of the length of the side opposite to \( \theta \) to the length of the hypotenuse, expressed as \( \sin(\theta) = \frac{opposite}{hypotenuse} \). Conversely, the cosine of \( \theta \) is the ratio of the length of the side adjacent to \( \theta \) to the length of the hypotenuse, written as \( \cos(\theta) = \frac{adjacent}{hypotenuse} \).

In the example exercise, through the inverse cosine function \( \cos^{-1} \), we find an angle whose cosine is known. Once this angle is established, we can use the sine function to find the ratio of the opposite side to the hypotenuse for this angle.

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in mathematics, particularly within geometry, concerning right triangles. It states that the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the other two sides \(a\) and \(b\). Formulated as \(a^2 + b^2 = c^2\), this theorem provides a relationship among all three sides of a right triangle.

In our exercise, the Pythagorean theorem plays a crucial role in step 3. After determining one of the triangle's sides as \(a = cx\), we use the Pythagorean theorem to find the length of the opposite side \(b\) by rearranging the formula and solving for \(b\). By doing so, we pave the way towards calculating the sine of the related angle.

Right Triangle Relationships

The right triangle's properties offer a framework through which we can explore trigonometric concepts. Each right triangle is characterized by a 90-degree angle and two acute angles that sum up to 90 degrees. The relationship between the angles and the lengths of their opposite and adjacent sides form the basis for trigonometric functions.

In relationship to our exercise, understanding that \( \cos(\alpha) = x \) means recognizing that \( \alpha \) is the angle in question and \(x\) is the cosine ratio of its adjacent side to the hypotenuse. When we solve for the sine of the same angle, \( \sin(\alpha) \), we're finding the ratio of the opposite side to the hypotenuse, showcasing the interconnectedness of sine and cosine in these relationships.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. These include, but aren't limited to, the reciprocal, quotient, and Pythagorean identities. In the context of the Pythagorean identities, one of the most well-known ones relates to sine and cosine: \(\sin^2(\theta) + \cos^2(\theta) = 1\).

In the case of the original exercise, once we find the sine function in terms of the angle whose cosine is known, we've effectively employed a Pythagorean identity. This step showcases that the sine squared plus the cosine squared of an angle equals to 1. Hence, by knowing the value of \(x\), which represents \(\cos(\alpha)\), and applying the identity, we can deduce that \(\sin(\alpha) = \sqrt{1 - x^2}\), thereby converting the trigonometric expression into an algebraic one.