Question

In Exercises 41 and \(42,\) cvaluate the expression. $$\sin \left(\cos ^{-1}\left(\frac{7}{11}\right)\right)$$

Short answer

\The result of expression \(\sin \left(\cos ^{-1}\left(\frac{7}{11}\right)\right) = \frac{8}{11}\)

Step-by-step solution

Calculate The Arc Cosine

Calculate the angle whose cosine gives \(7/11\), this is done through the arc cosine, or the inverse cosine function where \(\cos^{-1} \left( \frac{7}{11} \right)\). This gives a value of \(\theta\).

Understand the Relationship Between Sine and Cosine

In a right triangle, \(\sin(\theta) = \sqrt{1 - \cos^2(\theta)}\). This relationship comes from the Pythagorean Identity in trigonometry.

Substitute The Value of Cosine

We substitute the given value of cosine into the relationship to get our answer. That is, \(\sin \left( \cos^{-1} \left( \frac{7}{11} \right) \right) = \sin (\theta) = \sqrt{1 - \left( \frac{7}{11} \right)^2} = \sqrt{1 - \left( \frac{49}{121} \right)}\).

Simplify the Expression

Simplify the square root expression to get the final result. This becomes \( \sqrt{ \frac{121 - 49} {121} } = \sqrt{ \frac{72} {121} } = \frac{8} {11}\)

Key concepts

Arccosine Calculation

Understanding arccosine is essential when working with trigonometric functions. The arccosine function, often written as \(\cos^{-1}\) or \(\arccos\), is used to find the angle whose cosine is a given value. In the exercise provided, \(\cos^{-1}\left(\frac{7}{11}\right)\) essentially asks for the angle, which we denote as \(\theta\), whose cosine is \(\frac{7}{11}\).

To visualize this, imagine a right-angled triangle where the length of the adjacent side to \(\theta\) is 7 units and the hypotenuse is 11 units. The arccosine function in a calculator or a mathematical software can then be used to find this angle accurately. It's important to remember that the result will be in radians unless the calculator is set to degree mode.

Sine and Cosine Relationship

The sine and cosine functions are fundamental in trigonometry and are directly related to each other through the Pythagorean identity. For any angle \(\theta\), the sine and cosine functions satisfy the relationship \(\sin^2(\theta) + \cos^2(\theta) = 1\).

This comes from the Pythagorean theorem, which in the context of trigonometry, relates the sides of a right triangle to the functions of its angles. The relationship is pivotal in finding the value of one trigonometric function when given another. As an example, knowing the cosine of an angle allows us to find the sine of the angle by rearranging the Pythagorean identity to \(\sin(\theta) = \sqrt{1 - \cos^2(\theta)}\). This property was used in the exercise to find the sine of the angle when the cosine value \(\frac{7}{11}\) was known.

Pythagorean Identity in Trigonometry

The Pythagorean identity is a direct result of the Pythagorean theorem and is a pillar in trigonometry. The identity states that for any angle in a right triangle, the square of the sine plus the square of the cosine of the angle equals one: \(\sin^2(\theta) + \cos^2(\theta) = 1\).

This identity is a powerful tool when solving trigonometric equations as it allows us to express one trigonometric function solely in terms of another. In the context of the exercise, knowing the cosine of \(\theta\) allowed us to use the Pythagorean identity to determine the sine of \(\theta\), by substituting the known value into the identity and solving for the sine.

Simplifying Square Root Expressions

When dealing with trigonometry problems, simplifying square root expressions is a common task. In the final step of the exercise, the expression under the square root was simplified to isolate \(\sin(\theta)\).

The process involves subtracting the square of the cosine value from one, as shown by the Pythagorean identity, and then simplifying the resulting fraction inside the square root. This is done by finding the difference in the numerator and keeping the denominator intact, followed by taking the square root of both the numerator and denominator separately, if possible. Simplifying the square root of \(\frac{72}{121}\) results in \(\frac{8}{11}\), which is the precise value of \(\sin(\theta)\). Being adept at such simplifications can significantly streamline the process of solving trigonometric equations.